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\newtheorem{axm}{Axiom}[section]
\newtheorem{abr}{Abbreviation}[section]
\newtheorem{thm}{Theorem}[section]
\newtheorem{dec}{Rule Declaration}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}{Proposition}
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\title{Further Theorems of Propositional Calculus}
\author{Michael Meyling}
\date{\tt <module@qedeq.org>}


\begin{document}

\maketitle

\setlongtables



{\small
This document is part of the project ``Hilbert II''.
To get more information about this project look at:\\
\mbox{\url{http://www.qedeq.org}}.


Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no
Back-Cover Texts. See also under \url{http://www.gnu.org/copyleft/}
}


\section*{Abstract}

This module includes proofs of popositional calculus theorems. The following theorems and proofs are adapted from D. Hilbert and W. Ackermann's `Grundzuege der theoretischen Logik' (Berlin 1928, Springer)


\section*{Specification}

This document has the following specification:

\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & prophilbert3 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{http://www.qedeq.org/0_00_53/prophilbert3_1.00.00_1.00.00.qedeq} \\
\end{longtable}

\medskip

Author of this module: 
\begin{longtable}[h!]{l@{\extracolsep{\fill}}l}
Michael Meyling & mime@qedeq.org \\
\end{longtable}




\section*{References}

This document uses the results of the following documents:

\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & prophilbert2 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{prophilbert2_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{prophilbert2_1.00.00_1.00.00.pdf}  \\
\end{longtable}


\section*{Content}




First distributive law (first direction):




\begin{thm}[hilb36]
\hypertarget{hilb36}{}
\begin{displaymath}
(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb36:1}
  $1$ & $(P\ \wedge \  Q)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb24}{hilb24} } \\
\label{hilb36:2}
  $2$ & $(P\ \wedge \  A)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb36:1]{$1$}
} \\
\label{hilb36:3}
  $3$ & $(B\ \wedge \  A)\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb36:2]{$2$}
} \\
\label{hilb36:4}
  $4$ & $(Q\ \wedge \  A)\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb36:3]{$3$}
} \\
\label{hilb36:5}
  $5$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb36:6}
  $6$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb36:5]{$5$}
} \\
\label{hilb36:7}
  $7$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb36:6]{$6$}
} \\
\label{hilb36:8}
  $8$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb36:7]{$7$}
} \\
\label{hilb36:9}
  $9$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb36:8]{$8$}
} \\
\label{hilb36:10}
  $10$ & $(D\ \rightarrow \  Q)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q$ in \hyperref[hilb36:9]{$9$}
} \\
\label{hilb36:11}
  $11$ & $((Q\ \wedge \  A)\ \rightarrow \  Q)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \wedge \  A$ in \hyperref[hilb36:10]{$10$}
} \\
\label{hilb36:12}
  $12$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:4]{$4$}, \hyperref[hilb36:11]{$11$}} \\
\label{hilb36:13}
  $13$ & $(P\ \wedge \  Q)\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb25}{hilb25} } \\
\label{hilb36:14}
  $14$ & $(P\ \wedge \  A)\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb36:13]{$13$}
} \\
\label{hilb36:15}
  $15$ & $(B\ \wedge \  A)\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb36:14]{$14$}
} \\
\label{hilb36:16}
  $16$ & $(Q\ \wedge \  A)\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb36:15]{$15$}
} \\
\label{hilb36:17}
  $17$ & $(D\ \rightarrow \  A)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $A$ in \hyperref[hilb36:9]{$9$}
} \\
\label{hilb36:18}
  $18$ & $((Q\ \wedge \  A)\ \rightarrow \  A)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \wedge \  A$ in \hyperref[hilb36:17]{$17$}
} \\
\label{hilb36:19}
  $19$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:16]{$16$}, \hyperref[hilb36:18]{$18$}} \\
\label{hilb36:20}
  $20$ & $P\ \rightarrow \  (Q\ \rightarrow \  (P\ \wedge \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb28}{hilb28} } \\
\label{hilb36:21}
  $21$ & $P\ \rightarrow \  (A\ \rightarrow \  (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb36:20]{$20$}
} \\
\label{hilb36:22}
  $22$ & $B\ \rightarrow \  (A\ \rightarrow \  (B\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb36:21]{$21$}
} \\
\label{hilb36:23}
  $23$ & $B\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (B\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  A$ in \hyperref[hilb36:22]{$22$}
} \\
\label{hilb36:24}
  $24$ & $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb36:23]{$23$}
} \\
\label{hilb36:25}
  $25$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb36:26}
  $26$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb36:25]{$25$}
} \\
\label{hilb36:27}
  $27$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb36:26]{$26$}
} \\
\label{hilb36:28}
  $28$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb36:27]{$27$}
} \\
\label{hilb36:29}
  $29$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb36:28]{$28$}
} \\
\label{hilb36:30}
  $30$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb36:29]{$29$}
} \\
\label{hilb36:31}
  $31$ & $((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  Q$ in \hyperref[hilb36:30]{$30$}
} \\
\label{hilb36:32}
  $32$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:24]{$24$}, \hyperref[hilb36:31]{$31$}} \\
\label{hilb36:33}
  $33$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:12]{$12$}, \hyperref[hilb36:32]{$32$}} \\
\label{hilb36:34}
  $34$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (Q\ \rightarrow \  (P\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb16}{hilb16} } \\
\label{hilb36:35}
  $35$ & $(P\ \rightarrow \  (Q\ \rightarrow \  B))\ \rightarrow \  (Q\ \rightarrow \  (P\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb36:34]{$34$}
} \\
\label{hilb36:36}
  $36$ & $(P\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  (C\ \rightarrow \  (P\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb36:35]{$35$}
} \\
\label{hilb36:37}
  $37$ & $(D\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  (C\ \rightarrow \  (D\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb36:36]{$36$}
} \\
\label{hilb36:38}
  $38$ & $(D\ \rightarrow \  (C\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (C\ \rightarrow \  (D\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb36:37]{$37$}
} \\
\label{hilb36:39}
  $39$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (D\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb36:38]{$38$}
} \\
\label{hilb36:40}
  $40$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb36:39]{$39$}
} \\
\label{hilb36:41}
  $41$ & $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:33]{$33$}, \hyperref[hilb36:40]{$40$}} \\
\label{hilb36:42}
  $42$ & $(D\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb36:29]{$29$}
} \\
\label{hilb36:43}
  $43$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  A$ in \hyperref[hilb36:42]{$42$}
} \\
\label{hilb36:44}
  $44$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:41]{$41$}, \hyperref[hilb36:43]{$43$}} \\
\label{hilb36:45}
  $45$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:19]{$19$}, \hyperref[hilb36:44]{$44$}} \\
\label{hilb36:46}
  $46$ & $(P\ \rightarrow \  (P\ \rightarrow \  Q))\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb33}{hilb33} } \\
\label{hilb36:47}
  $47$ & $(P\ \rightarrow \  (P\ \rightarrow \  A))\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb36:46]{$46$}
} \\
\label{hilb36:48}
  $48$ & $(B\ \rightarrow \  (B\ \rightarrow \  A))\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb36:47]{$47$}
} \\
\label{hilb36:49}
  $49$ & $(B\ \rightarrow \  (B\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (B\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb36:48]{$48$}
} \\
\label{hilb36:50}
  $50$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb36:49]{$49$}
} \\
\label{hilb36:51}
  $51$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb36:45]{$45$}, \hyperref[hilb36:50]{$50$}} \\
 & & \qedhere
\end{longtable}
\end{proof}



First distributive law (second direction):




\begin{thm}[hilb37]
\hypertarget{hilb37}{}
\begin{displaymath}
((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb37:1}
  $1$ & $P\ \rightarrow \  (Q\ \rightarrow \  (P\ \wedge \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb28}{hilb28} } \\
\label{hilb37:2}
  $2$ & $P\ \rightarrow \  (A\ \rightarrow \  (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb37:1]{$1$}
} \\
\label{hilb37:3}
  $3$ & $B\ \rightarrow \  (A\ \rightarrow \  (B\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb37:2]{$2$}
} \\
\label{hilb37:4}
  $4$ & $Q\ \rightarrow \  (A\ \rightarrow \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb37:3]{$3$}
} \\
\label{hilb37:5}
  $5$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb37:6}
  $6$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb37:5]{$5$}
} \\
\label{hilb37:7}
  $7$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb37:6]{$6$}
} \\
\label{hilb37:8}
  $8$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb37:7]{$7$}
} \\
\label{hilb37:9}
  $9$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb37:8]{$8$}
} \\
\label{hilb37:10}
  $10$ & $(D\ \rightarrow \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q\ \wedge \  A$ in \hyperref[hilb37:9]{$9$}
} \\
\label{hilb37:11}
  $11$ & $(A\ \rightarrow \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A$ in \hyperref[hilb37:10]{$10$}
} \\
\label{hilb37:12}
  $12$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb37:13}
  $13$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb37:12]{$12$}
} \\
\label{hilb37:14}
  $14$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb37:13]{$13$}
} \\
\label{hilb37:15}
  $15$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb37:14]{$14$}
} \\
\label{hilb37:16}
  $16$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \rightarrow \  D)\ \rightarrow \  (Q\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:17}
  $17$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((Q\ \rightarrow \  D)\ \rightarrow \  (Q\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:16]{$16$}
} \\
\label{hilb37:18}
  $18$ & $((A\ \rightarrow \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((Q\ \rightarrow \  (A\ \rightarrow \  (Q\ \wedge \  A)))\ \rightarrow \  (Q\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $A\ \rightarrow \  (Q\ \wedge \  A)$ in \hyperref[hilb37:17]{$17$}
} \\
\label{hilb37:19}
  $19$ & $(Q\ \rightarrow \  (A\ \rightarrow \  (Q\ \wedge \  A)))\ \rightarrow \  (Q\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:11]{$11$}, \hyperref[hilb37:18]{$18$}} \\
\label{hilb37:20}
  $20$ & $Q\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:4]{$4$}, \hyperref[hilb37:19]{$19$}} \\
\label{hilb37:21}
  $21$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  (Q\ \rightarrow \  (P\ \rightarrow \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb16}{hilb16} } \\
\label{hilb37:22}
  $22$ & $(P\ \rightarrow \  (Q\ \rightarrow \  B))\ \rightarrow \  (Q\ \rightarrow \  (P\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb37:21]{$21$}
} \\
\label{hilb37:23}
  $23$ & $(P\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  (C\ \rightarrow \  (P\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb37:22]{$22$}
} \\
\label{hilb37:24}
  $24$ & $(D\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  (C\ \rightarrow \  (D\ \rightarrow \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb37:23]{$23$}
} \\
\label{hilb37:25}
  $25$ & $(D\ \rightarrow \  (C\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (C\ \rightarrow \  (D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:24]{$24$}
} \\
\label{hilb37:26}
  $26$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:25]{$25$}
} \\
\label{hilb37:27}
  $27$ & $(Q\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q$ in \hyperref[hilb37:26]{$26$}
} \\
\label{hilb37:28}
  $28$ & $(P\ \vee \  A)\ \rightarrow \  (Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:20]{$20$}, \hyperref[hilb37:27]{$27$}} \\
\label{hilb37:29}
  $29$ & $(D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:9]{$9$}
} \\
\label{hilb37:30}
  $30$ & $(Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q$ in \hyperref[hilb37:29]{$29$}
} \\
\label{hilb37:31}
  $31$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:32}
  $32$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:31]{$31$}
} \\
\label{hilb37:33}
  $33$ & $((Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  (Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:32]{$32$}
} \\
\label{hilb37:34}
  $34$ & $((P\ \vee \  A)\ \rightarrow \  (Q\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:30]{$30$}, \hyperref[hilb37:33]{$33$}} \\
\label{hilb37:35}
  $35$ & $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:28]{$28$}, \hyperref[hilb37:34]{$34$}} \\
\label{hilb37:36}
  $36$ & $(P\ \vee \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb14}{hilb14} } \\
\label{hilb37:37}
  $37$ & $(P\ \vee \  (Q\ \vee \  B))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb37:36]{$36$}
} \\
\label{hilb37:38}
  $38$ & $(P\ \vee \  (C\ \vee \  B))\ \rightarrow \  ((P\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb37:37]{$37$}
} \\
\label{hilb37:39}
  $39$ & $(D\ \vee \  (C\ \vee \  B))\ \rightarrow \  ((D\ \vee \  C)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb37:38]{$38$}
} \\
\label{hilb37:40}
  $40$ & $(D\ \vee \  (C\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((D\ \vee \  C)\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  A$ in \hyperref[hilb37:39]{$39$}
} \\
\label{hilb37:41}
  $41$ & $(D\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((D\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb37:40]{$40$}
} \\
\label{hilb37:42}
  $42$ & $(P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P$ in \hyperref[hilb37:41]{$41$}
} \\
\label{hilb37:43}
  $43$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\
\label{hilb37:44}
  $44$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\
\label{hilb37:45}
  $45$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb37:8]{$8$}
} \\
\label{hilb37:46}
  $46$ & $(D\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:45]{$45$}
} \\
\label{hilb37:47}
  $47$ & $((P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:46]{$46$}
} \\
\label{hilb37:48}
  $48$ & $(\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:42]{$42$}, \hyperref[hilb37:47]{$47$}} \\
\label{hilb37:49}
  $49$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb37:43]{$43$}
} \\
\label{hilb37:50}
  $50$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb37:49]{$49$}
} \\
\label{hilb37:51}
  $51$ & $(C\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg C\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:50]{$50$}
} \\
\label{hilb37:52}
  $52$ & $((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb37:51]{$51$}
} \\
\label{hilb37:53}
  $53$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:54}
  $54$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:53]{$53$}
} \\
\label{hilb37:55}
  $55$ & $((\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:54]{$54$}
} \\
\label{hilb37:56}
  $56$ & $(((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:48]{$48$}, \hyperref[hilb37:55]{$55$}} \\
\label{hilb37:57}
  $57$ & $((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:52]{$52$}, \hyperref[hilb37:56]{$56$}} \\
\label{hilb37:58}
  $58$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb37:44]{$44$}
} \\
\label{hilb37:59}
  $59$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb37:58]{$58$}
} \\
\label{hilb37:60}
  $60$ & $(\neg C\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (C\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:59]{$59$}
} \\
\label{hilb37:61}
  $61$ & $(\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb37:60]{$60$}
} \\
\label{hilb37:62}
  $62$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:53]{$53$}
} \\
\label{hilb37:63}
  $63$ & $((\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:62]{$62$}
} \\
\label{hilb37:64}
  $64$ & $(((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:61]{$61$}, \hyperref[hilb37:63]{$63$}} \\
\label{hilb37:65}
  $65$ & $((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:57]{$57$}, \hyperref[hilb37:64]{$64$}} \\
\label{hilb37:66}
  $66$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb37:8]{$8$}
} \\
\label{hilb37:67}
  $67$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:66]{$66$}
} \\
\label{hilb37:68}
  $68$ & $(((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:67]{$67$}
} \\
\label{hilb37:69}
  $69$ & $(\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:65]{$65$}, \hyperref[hilb37:68]{$68$}} \\
\label{hilb37:70}
  $70$ & $(C\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg C\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:50]{$50$}
} \\
\label{hilb37:71}
  $71$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:70]{$70$}
} \\
\label{hilb37:72}
  $72$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:73}
  $73$ & $(D\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:72]{$72$}
} \\
\label{hilb37:74}
  $74$ & $((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$ in \hyperref[hilb37:73]{$73$}
} \\
\label{hilb37:75}
  $75$ & $(((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:69]{$69$}, \hyperref[hilb37:74]{$74$}} \\
\label{hilb37:76}
  $76$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:71]{$71$}, \hyperref[hilb37:75]{$75$}} \\
\label{hilb37:77}
  $77$ & $(\neg C\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (C\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:59]{$59$}
} \\
\label{hilb37:78}
  $78$ & $(\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:77]{$77$}
} \\
\label{hilb37:79}
  $79$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:72]{$72$}
} \\
\label{hilb37:80}
  $80$ & $((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:79]{$79$}
} \\
\label{hilb37:81}
  $81$ & $(((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:78]{$78$}, \hyperref[hilb37:80]{$80$}} \\
\label{hilb37:82}
  $82$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:76]{$76$}, \hyperref[hilb37:81]{$81$}} \\
\label{hilb37:83}
  $83$ & $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:35]{$35$}, \hyperref[hilb37:82]{$82$}} \\
\label{hilb37:84}
  $84$ & $(P\ \vee \  P)\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb11}{hilb11} } \\
\label{hilb37:85}
  $85$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  D)\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  A$ in \hyperref[hilb37:8]{$8$}
} \\
\label{hilb37:86}
  $86$ & $(D\ \rightarrow \  P)\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  D)\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb37:85]{$85$}
} \\
\label{hilb37:87}
  $87$ & $((P\ \vee \  P)\ \rightarrow \  P)\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  P$ in \hyperref[hilb37:86]{$86$}
} \\
\label{hilb37:88}
  $88$ & $((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:84]{$84$}, \hyperref[hilb37:87]{$87$}} \\
\label{hilb37:89}
  $89$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{hilb37:90}
  $90$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb37:89]{$89$}
} \\
\label{hilb37:91}
  $91$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb37:90]{$90$}
} \\
\label{hilb37:92}
  $92$ & $(C\ \vee \  P)\ \rightarrow \  (P\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb37:91]{$91$}
} \\
\label{hilb37:93}
  $93$ & $((Q\ \wedge \  A)\ \vee \  P)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q\ \wedge \  A$ in \hyperref[hilb37:92]{$92$}
} \\
\label{hilb37:94}
  $94$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  D)\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(Q\ \wedge \  A)\ \vee \  (P\ \vee \  P)$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:95}
  $95$ & $(D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  D)\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:94]{$94$}
} \\
\label{hilb37:96}
  $96$ & $(((Q\ \wedge \  A)\ \vee \  P)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  P))\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(Q\ \wedge \  A)\ \vee \  P$ in \hyperref[hilb37:95]{$95$}
} \\
\label{hilb37:97}
  $97$ & $(((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  P))\ \rightarrow \  (((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:93]{$93$}, \hyperref[hilb37:96]{$96$}} \\
\label{hilb37:98}
  $98$ & $((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:88]{$88$}, \hyperref[hilb37:97]{$97$}} \\
\label{hilb37:99}
  $99$ & $(C\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  A$ in \hyperref[hilb37:91]{$91$}
} \\
\label{hilb37:100}
  $100$ & $((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  P$ in \hyperref[hilb37:99]{$99$}
} \\
\label{hilb37:101}
  $101$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:102}
  $102$ & $(D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:101]{$101$}
} \\
\label{hilb37:103}
  $103$ & $(((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P)))\ \rightarrow \  (((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(Q\ \wedge \  A)\ \vee \  (P\ \vee \  P)$ in \hyperref[hilb37:102]{$102$}
} \\
\label{hilb37:104}
  $104$ & $(((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((Q\ \wedge \  A)\ \vee \  (P\ \vee \  P)))\ \rightarrow \  (((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:98]{$98$}, \hyperref[hilb37:103]{$103$}} \\
\label{hilb37:105}
  $105$ & $((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:100]{$100$}, \hyperref[hilb37:104]{$104$}} \\
\label{hilb37:106}
  $106$ & $(D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:45]{$45$}
} \\
\label{hilb37:107}
  $107$ & $(((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:106]{$106$}
} \\
\label{hilb37:108}
  $108$ & $(\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:105]{$105$}, \hyperref[hilb37:107]{$107$}} \\
\label{hilb37:109}
  $109$ & $(C\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg C\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:50]{$50$}
} \\
\label{hilb37:110}
  $110$ & $((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb37:109]{$109$}
} \\
\label{hilb37:111}
  $111$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:112}
  $112$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:111]{$111$}
} \\
\label{hilb37:113}
  $113$ & $((\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:112]{$112$}
} \\
\label{hilb37:114}
  $114$ & $(((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:108]{$108$}, \hyperref[hilb37:113]{$113$}} \\
\label{hilb37:115}
  $115$ & $((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:110]{$110$}, \hyperref[hilb37:114]{$114$}} \\
\label{hilb37:116}
  $116$ & $(\neg C\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (C\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:59]{$59$}
} \\
\label{hilb37:117}
  $117$ & $(\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb37:116]{$116$}
} \\
\label{hilb37:118}
  $118$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:111]{$111$}
} \\
\label{hilb37:119}
  $119$ & $((\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:118]{$118$}
} \\
\label{hilb37:120}
  $120$ & $(((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:117]{$117$}, \hyperref[hilb37:119]{$119$}} \\
\label{hilb37:121}
  $121$ & $((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:115]{$115$}, \hyperref[hilb37:120]{$120$}} \\
\label{hilb37:122}
  $122$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:66]{$66$}
} \\
\label{hilb37:123}
  $123$ & $(((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:122]{$122$}
} \\
\label{hilb37:124}
  $124$ & $(\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:121]{$121$}, \hyperref[hilb37:123]{$123$}} \\
\label{hilb37:125}
  $125$ & $(C\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg C\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:50]{$50$}
} \\
\label{hilb37:126}
  $126$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:125]{$125$}
} \\
\label{hilb37:127}
  $127$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:15]{$15$}
} \\
\label{hilb37:128}
  $128$ & $(D\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:127]{$127$}
} \\
\label{hilb37:129}
  $129$ & $((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:128]{$128$}
} \\
\label{hilb37:130}
  $130$ & $(((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:124]{$124$}, \hyperref[hilb37:129]{$129$}} \\
\label{hilb37:131}
  $131$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:126]{$126$}, \hyperref[hilb37:130]{$130$}} \\
\label{hilb37:132}
  $132$ & $(\neg C\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (C\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb37:59]{$59$}
} \\
\label{hilb37:133}
  $133$ & $(\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:132]{$132$}
} \\
\label{hilb37:134}
  $134$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:127]{$127$}
} \\
\label{hilb37:135}
  $135$ & $((\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  ((((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$ in \hyperref[hilb37:134]{$134$}
} \\
\label{hilb37:136}
  $136$ & $(((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))\ \rightarrow \  (((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:133]{$133$}, \hyperref[hilb37:135]{$135$}} \\
\label{hilb37:137}
  $137$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  P)\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:131]{$131$}, \hyperref[hilb37:136]{$136$}} \\
\label{hilb37:138}
  $138$ & $(P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:83]{$83$}, \hyperref[hilb37:137]{$137$}} \\
\label{hilb37:139}
  $139$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (D\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb37:25]{$25$}
} \\
\label{hilb37:140}
  $140$ & $((P\ \vee \  A)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  A$ in \hyperref[hilb37:139]{$139$}
} \\
\label{hilb37:141}
  $141$ & $(P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:138]{$138$}, \hyperref[hilb37:140]{$140$}} \\
\label{hilb37:142}
  $142$ & $(P\ \rightarrow \  (Q\ \rightarrow \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert2_1.00.00_1.00.00.pdf}{}{hilb29}{hilb29} } \\
\label{hilb37:143}
  $143$ & $(P\ \rightarrow \  (Q\ \rightarrow \  B))\ \rightarrow \  ((P\ \wedge \  Q)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb37:142]{$142$}
} \\
\label{hilb37:144}
  $144$ & $(P\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  ((P\ \wedge \  C)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb37:143]{$143$}
} \\
\label{hilb37:145}
  $145$ & $(D\ \rightarrow \  (C\ \rightarrow \  B))\ \rightarrow \  ((D\ \wedge \  C)\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb37:144]{$144$}
} \\
\label{hilb37:146}
  $146$ & $(D\ \rightarrow \  (C\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((D\ \wedge \  C)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb37:145]{$145$}
} \\
\label{hilb37:147}
  $147$ & $(D\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((D\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb37:146]{$146$}
} \\
\label{hilb37:148}
  $148$ & $((P\ \vee \  Q)\ \rightarrow \  ((P\ \vee \  A)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \vee \  Q$ in \hyperref[hilb37:147]{$147$}
} \\
\label{hilb37:149}
  $149$ & $((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb37:141]{$141$}, \hyperref[hilb37:148]{$148$}} \\
 & & \qedhere
\end{longtable}
\end{proof}



A form for the abbreviation rule form for disjunction (first direction):




\begin{thm}[hilb38]
\hypertarget{hilb38}{}
\begin{displaymath}
(P\ \vee \  Q)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb38:1}
  $1$ & $P\ \rightarrow \  \neg \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\
\label{hilb38:2}
  $2$ & $A\ \rightarrow \  \neg \neg A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb38:1]{$1$}
} \\
\label{hilb38:3}
  $3$ & $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb38:2]{$2$}
} \\
\label{hilb38:4}
  $4$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\
\label{hilb38:5}
  $5$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\
\label{hilb38:6}
  $6$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb38:4]{$4$}
} \\
\label{hilb38:7}
  $7$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb38:6]{$6$}
} \\
\label{hilb38:8}
  $8$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb38:9}
  $9$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb38:8]{$8$}
} \\
\label{hilb38:10}
  $10$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb38:9]{$9$}
} \\
\label{hilb38:11}
  $11$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb38:10]{$10$}
} \\
\label{hilb38:12}
  $12$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb38:5]{$5$}
} \\
\label{hilb38:13}
  $13$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb38:12]{$12$}
} \\
\label{hilb38:14}
  $14$ & $Q\ \rightarrow \  \neg \neg Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb38:1]{$1$}
} \\
\label{hilb38:15}
  $15$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb38:16}
  $16$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb38:15]{$15$}
} \\
\label{hilb38:17}
  $17$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb38:16]{$16$}
} \\
\label{hilb38:18}
  $18$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb38:17]{$17$}
} \\
\label{hilb38:19}
  $19$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb38:18]{$18$}
} \\
\label{hilb38:20}
  $20$ & $(D\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  \neg \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P$ in \hyperref[hilb38:19]{$19$}
} \\
\label{hilb38:21}
  $21$ & $(P\ \rightarrow \  \neg \neg P)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P$ in \hyperref[hilb38:20]{$20$}
} \\
\label{hilb38:22}
  $22$ & $(Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:1]{$1$}, \hyperref[hilb38:21]{$21$}} \\
\label{hilb38:23}
  $23$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{hilb38:24}
  $24$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb38:23]{$23$}
} \\
\label{hilb38:25}
  $25$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb38:24]{$24$}
} \\
\label{hilb38:26}
  $26$ & $(B\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P$ in \hyperref[hilb38:25]{$25$}
} \\
\label{hilb38:27}
  $27$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb38:26]{$26$}
} \\
\label{hilb38:28}
  $28$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  P$ in \hyperref[hilb38:11]{$11$}
} \\
\label{hilb38:29}
  $29$ & $(D\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb38:28]{$28$}
} \\
\label{hilb38:30}
  $30$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb38:29]{$29$}
} \\
\label{hilb38:31}
  $31$ & $((Q\ \vee \  P)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:27]{$27$}, \hyperref[hilb38:30]{$30$}} \\
\label{hilb38:32}
  $32$ & $(Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:22]{$22$}, \hyperref[hilb38:31]{$31$}} \\
\label{hilb38:33}
  $33$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:11]{$11$}
} \\
\label{hilb38:34}
  $34$ & $(D\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb38:33]{$33$}
} \\
\label{hilb38:35}
  $35$ & $((Q\ \vee \  P)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  P$ in \hyperref[hilb38:34]{$34$}
} \\
\label{hilb38:36}
  $36$ & $((P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:32]{$32$}, \hyperref[hilb38:35]{$35$}} \\
\label{hilb38:37}
  $37$ & $(P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:23]{$23$}, \hyperref[hilb38:36]{$36$}} \\
\label{hilb38:38}
  $38$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\
\label{hilb38:39}
  $39$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb38:38]{$38$}
} \\
\label{hilb38:40}
  $40$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb38:39]{$39$}
} \\
\label{hilb38:41}
  $41$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb38:40]{$40$}
} \\
\label{hilb38:42}
  $42$ & $((P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:41]{$41$}
} \\
\label{hilb38:43}
  $43$ & $\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:37]{$37$}, \hyperref[hilb38:42]{$42$}} \\
\label{hilb38:44}
  $44$ & $(B\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb38:40]{$40$}
} \\
\label{hilb38:45}
  $45$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:44]{$44$}
} \\
\label{hilb38:46}
  $46$ & $\neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:43]{$43$}, \hyperref[hilb38:45]{$45$}} \\
\label{hilb38:47}
  $47$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb38:18]{$18$}
} \\
\label{hilb38:48}
  $48$ & $(D\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:47]{$47$}
} \\
\label{hilb38:49}
  $49$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb38:48]{$48$}
} \\
\label{hilb38:50}
  $50$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:46]{$46$}, \hyperref[hilb38:49]{$49$}} \\
\label{hilb38:51}
  $51$ & $(B\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg B\ \vee \  \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb38:7]{$7$}
} \\
\label{hilb38:52}
  $52$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:51]{$51$}
} \\
\label{hilb38:53}
  $53$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb38:11]{$11$}
} \\
\label{hilb38:54}
  $54$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:53]{$53$}
} \\
\label{hilb38:55}
  $55$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb38:54]{$54$}
} \\
\label{hilb38:56}
  $56$ & $(((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:50]{$50$}, \hyperref[hilb38:55]{$55$}} \\
\label{hilb38:57}
  $57$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:52]{$52$}, \hyperref[hilb38:56]{$56$}} \\
\label{hilb38:58}
  $58$ & $(\neg B\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (B\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:13]{$13$}
} \\
\label{hilb38:59}
  $59$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:58]{$58$}
} \\
\label{hilb38:60}
  $60$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:53]{$53$}
} \\
\label{hilb38:61}
  $61$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:60]{$60$}
} \\
\label{hilb38:62}
  $62$ & $(((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:59]{$59$}, \hyperref[hilb38:61]{$61$}} \\
\label{hilb38:63}
  $63$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:57]{$57$}, \hyperref[hilb38:62]{$62$}} \\
\label{hilb38:64}
  $64$ & $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:3]{$3$}, \hyperref[hilb38:63]{$63$}} \\
\label{hilb38:65}
  $65$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb38:18]{$18$}
} \\
\label{hilb38:66}
  $66$ & $(D\ \rightarrow \  \neg \neg Q)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg Q$ in \hyperref[hilb38:65]{$65$}
} \\
\label{hilb38:67}
  $67$ & $(Q\ \rightarrow \  \neg \neg Q)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q$ in \hyperref[hilb38:66]{$66$}
} \\
\label{hilb38:68}
  $68$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:14]{$14$}, \hyperref[hilb38:67]{$67$}} \\
\label{hilb38:69}
  $69$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  \neg \neg Q$ in \hyperref[hilb38:40]{$40$}
} \\
\label{hilb38:70}
  $70$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb38:69]{$69$}
} \\
\label{hilb38:71}
  $71$ & $\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:68]{$68$}, \hyperref[hilb38:70]{$70$}} \\
\label{hilb38:72}
  $72$ & $(B\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:40]{$40$}
} \\
\label{hilb38:73}
  $73$ & $(\neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:72]{$72$}
} \\
\label{hilb38:74}
  $74$ & $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:71]{$71$}, \hyperref[hilb38:73]{$73$}} \\
\label{hilb38:75}
  $75$ & $(D\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:47]{$47$}
} \\
\label{hilb38:76}
  $76$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:75]{$75$}
} \\
\label{hilb38:77}
  $77$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:74]{$74$}, \hyperref[hilb38:76]{$76$}} \\
\label{hilb38:78}
  $78$ & $(B\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg B\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:7]{$7$}
} \\
\label{hilb38:79}
  $79$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:78]{$78$}
} \\
\label{hilb38:80}
  $80$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:11]{$11$}
} \\
\label{hilb38:81}
  $81$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:80]{$80$}
} \\
\label{hilb38:82}
  $82$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb38:81]{$81$}
} \\
\label{hilb38:83}
  $83$ & $(((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:77]{$77$}, \hyperref[hilb38:82]{$82$}} \\
\label{hilb38:84}
  $84$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:79]{$79$}, \hyperref[hilb38:83]{$83$}} \\
\label{hilb38:85}
  $85$ & $(\neg B\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (B\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:13]{$13$}
} \\
\label{hilb38:86}
  $86$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb38:85]{$85$}
} \\
\label{hilb38:87}
  $87$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:80]{$80$}
} \\
\label{hilb38:88}
  $88$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb38:87]{$87$}
} \\
\label{hilb38:89}
  $89$ & $(((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:86]{$86$}, \hyperref[hilb38:88]{$88$}} \\
\label{hilb38:90}
  $90$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:84]{$84$}, \hyperref[hilb38:89]{$89$}} \\
\label{hilb38:91}
  $91$ & $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb38:64]{$64$}, \hyperref[hilb38:90]{$90$}} \\
\label{hilb38:92}
  $92$ & $(P\ \vee \  Q)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb38:91]{$91$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



A form for the abbreviation rule form for disjunction (second direction):




\begin{thm}[hilb39]
\hypertarget{hilb39}{}
\begin{displaymath}
\neg (\neg P\ \wedge \  \neg Q)\ \rightarrow \  (P\ \vee \  Q)\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb39:1}
  $1$ & $\neg \neg P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\
\label{hilb39:2}
  $2$ & $\neg \neg A\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb39:1]{$1$}
} \\
\label{hilb39:3}
  $3$ & $\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb39:2]{$2$}
} \\
\label{hilb39:4}
  $4$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\
\label{hilb39:5}
  $5$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\
\label{hilb39:6}
  $6$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{hilb39:7}
  $7$ & $(P\ \vee \  A)\ \rightarrow \  (A\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb39:6]{$6$}
} \\
\label{hilb39:8}
  $8$ & $(B\ \vee \  A)\ \rightarrow \  (A\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb39:7]{$7$}
} \\
\label{hilb39:9}
  $9$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb39:10}
  $10$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb39:9]{$9$}
} \\
\label{hilb39:11}
  $11$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb39:10]{$10$}
} \\
\label{hilb39:12}
  $12$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb39:11]{$11$}
} \\
\label{hilb39:13}
  $13$ & $(B\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb39:8]{$8$}
} \\
\label{hilb39:14}
  $14$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb39:4]{$4$}
} \\
\label{hilb39:15}
  $15$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg B\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb39:14]{$14$}
} \\
\label{hilb39:16}
  $16$ & $(B\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg B\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb39:15]{$15$}
} \\
\label{hilb39:17}
  $17$ & $(\neg P\ \vee \  A)\ \rightarrow \  (P\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb39:5]{$5$}
} \\
\label{hilb39:18}
  $18$ & $(\neg B\ \vee \  A)\ \rightarrow \  (B\ \rightarrow \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb39:17]{$17$}
} \\
\label{hilb39:19}
  $19$ & $(\neg B\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (B\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb39:18]{$18$}
} \\
\label{hilb39:20}
  $20$ & $\neg \neg Q\ \rightarrow \  Q$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $Q$ in \hyperref[hilb39:1]{$1$}
} \\
\label{hilb39:21}
  $21$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb39:22}
  $22$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb39:21]{$21$}
} \\
\label{hilb39:23}
  $23$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb39:22]{$22$}
} \\
\label{hilb39:24}
  $24$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb39:23]{$23$}
} \\
\label{hilb39:25}
  $25$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb39:24]{$24$}
} \\
\label{hilb39:26}
  $26$ & $(D\ \rightarrow \  P)\ \rightarrow \  ((Q\ \vee \  D)\ \rightarrow \  (Q\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb39:25]{$25$}
} \\
\label{hilb39:27}
  $27$ & $(\neg \neg P\ \rightarrow \  P)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg P$ in \hyperref[hilb39:26]{$26$}
} \\
\label{hilb39:28}
  $28$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:1]{$1$}, \hyperref[hilb39:27]{$27$}} \\
\label{hilb39:29}
  $29$ & $(B\ \vee \  P)\ \rightarrow \  (P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P$ in \hyperref[hilb39:8]{$8$}
} \\
\label{hilb39:30}
  $30$ & $(Q\ \vee \  P)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q$ in \hyperref[hilb39:29]{$29$}
} \\
\label{hilb39:31}
  $31$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:32}
  $32$ & $(D\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  D)\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb39:31]{$31$}
} \\
\label{hilb39:33}
  $33$ & $((Q\ \vee \  P)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  P$ in \hyperref[hilb39:32]{$32$}
} \\
\label{hilb39:34}
  $34$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (Q\ \vee \  P))\ \rightarrow \  ((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:30]{$30$}, \hyperref[hilb39:33]{$33$}} \\
\label{hilb39:35}
  $35$ & $(Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:28]{$28$}, \hyperref[hilb39:34]{$34$}} \\
\label{hilb39:36}
  $36$ & $(B\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $Q$ in \hyperref[hilb39:8]{$8$}
} \\
\label{hilb39:37}
  $37$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb39:36]{$36$}
} \\
\label{hilb39:38}
  $38$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:39}
  $39$ & $(D\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb39:38]{$38$}
} \\
\label{hilb39:40}
  $40$ & $((Q\ \vee \  \neg \neg P)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (((\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \vee \  \neg \neg P$ in \hyperref[hilb39:39]{$39$}
} \\
\label{hilb39:41}
  $41$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  \neg \neg P))\ \rightarrow \  ((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:35]{$35$}, \hyperref[hilb39:40]{$40$}} \\
\label{hilb39:42}
  $42$ & $(\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:37]{$37$}, \hyperref[hilb39:41]{$41$}} \\
\label{hilb39:43}
  $43$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\
\label{hilb39:44}
  $44$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb39:43]{$43$}
} \\
\label{hilb39:45}
  $45$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb39:44]{$44$}
} \\
\label{hilb39:46}
  $46$ & $(B\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:47}
  $47$ & $((\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb39:46]{$46$}
} \\
\label{hilb39:48}
  $48$ & $\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:42]{$42$}, \hyperref[hilb39:47]{$47$}} \\
\label{hilb39:49}
  $49$ & $(B\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:50}
  $50$ & $(\neg (P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb39:49]{$49$}
} \\
\label{hilb39:51}
  $51$ & $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:48]{$48$}, \hyperref[hilb39:50]{$50$}} \\
\label{hilb39:52}
  $52$ & $(B\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:53}
  $53$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:52]{$52$}
} \\
\label{hilb39:54}
  $54$ & $\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:51]{$51$}, \hyperref[hilb39:53]{$53$}} \\
\label{hilb39:55}
  $55$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb39:24]{$24$}
} \\
\label{hilb39:56}
  $56$ & $(D\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:55]{$55$}
} \\
\label{hilb39:57}
  $57$ & $(\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:56]{$56$}
} \\
\label{hilb39:58}
  $58$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:54]{$54$}, \hyperref[hilb39:57]{$57$}} \\
\label{hilb39:59}
  $59$ & $(B\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:8]{$8$}
} \\
\label{hilb39:60}
  $60$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb39:59]{$59$}
} \\
\label{hilb39:61}
  $61$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:62}
  $62$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:61]{$61$}
} \\
\label{hilb39:63}
  $63$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:62]{$62$}
} \\
\label{hilb39:64}
  $64$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:60]{$60$}, \hyperref[hilb39:63]{$63$}} \\
\label{hilb39:65}
  $65$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:58]{$58$}, \hyperref[hilb39:64]{$64$}} \\
\label{hilb39:66}
  $66$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:13]{$13$}
} \\
\label{hilb39:67}
  $67$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:68}
  $68$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:67]{$67$}
} \\
\label{hilb39:69}
  $69$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:68]{$68$}
} \\
\label{hilb39:70}
  $70$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:65]{$65$}, \hyperref[hilb39:69]{$69$}} \\
\label{hilb39:71}
  $71$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:66]{$66$}, \hyperref[hilb39:70]{$70$}} \\
\label{hilb39:72}
  $72$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb39:16]{$16$}
} \\
\label{hilb39:73}
  $73$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:74}
  $74$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:73]{$73$}
} \\
\label{hilb39:75}
  $75$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:74]{$74$}
} \\
\label{hilb39:76}
  $76$ & $((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:71]{$71$}, \hyperref[hilb39:75]{$75$}} \\
\label{hilb39:77}
  $77$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:72]{$72$}, \hyperref[hilb39:76]{$76$}} \\
\label{hilb39:78}
  $78$ & $(\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:19]{$19$}
} \\
\label{hilb39:79}
  $79$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$ in \hyperref[hilb39:73]{$73$}
} \\
\label{hilb39:80}
  $80$ & $((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:79]{$79$}
} \\
\label{hilb39:81}
  $81$ & $((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:78]{$78$}, \hyperref[hilb39:80]{$80$}} \\
\label{hilb39:82}
  $82$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:77]{$77$}, \hyperref[hilb39:81]{$81$}} \\
\label{hilb39:83}
  $83$ & $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:3]{$3$}, \hyperref[hilb39:82]{$82$}} \\
\label{hilb39:84}
  $84$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P$ in \hyperref[hilb39:24]{$24$}
} \\
\label{hilb39:85}
  $85$ & $(D\ \rightarrow \  Q)\ \rightarrow \  ((\neg \neg P\ \vee \  D)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q$ in \hyperref[hilb39:84]{$84$}
} \\
\label{hilb39:86}
  $86$ & $(\neg \neg Q\ \rightarrow \  Q)\ \rightarrow \  ((\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg Q$ in \hyperref[hilb39:85]{$85$}
} \\
\label{hilb39:87}
  $87$ & $(\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:20]{$20$}, \hyperref[hilb39:86]{$86$}} \\
\label{hilb39:88}
  $88$ & $(B\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg P\ \vee \  Q$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:89}
  $89$ & $((\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg P\ \vee \  \neg \neg Q$ in \hyperref[hilb39:88]{$88$}
} \\
\label{hilb39:90}
  $90$ & $\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:87]{$87$}, \hyperref[hilb39:89]{$89$}} \\
\label{hilb39:91}
  $91$ & $(B\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:92}
  $92$ & $(\neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:91]{$91$}
} \\
\label{hilb39:93}
  $93$ & $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:90]{$90$}, \hyperref[hilb39:92]{$92$}} \\
\label{hilb39:94}
  $94$ & $(B\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:45]{$45$}
} \\
\label{hilb39:95}
  $95$ & $(\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:94]{$94$}
} \\
\label{hilb39:96}
  $96$ & $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:93]{$93$}, \hyperref[hilb39:95]{$95$}} \\
\label{hilb39:97}
  $97$ & $(D\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:55]{$55$}
} \\
\label{hilb39:98}
  $98$ & $(\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:97]{$97$}
} \\
\label{hilb39:99}
  $99$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:96]{$96$}, \hyperref[hilb39:98]{$98$}} \\
\label{hilb39:100}
  $100$ & $(B\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:8]{$8$}
} \\
\label{hilb39:101}
  $101$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb39:100]{$100$}
} \\
\label{hilb39:102}
  $102$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:103}
  $103$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:102]{$102$}
} \\
\label{hilb39:104}
  $104$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:103]{$103$}
} \\
\label{hilb39:105}
  $105$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)))\ \rightarrow \  (((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:101]{$101$}, \hyperref[hilb39:104]{$104$}} \\
\label{hilb39:106}
  $106$ & $((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:99]{$99$}, \hyperref[hilb39:105]{$105$}} \\
\label{hilb39:107}
  $107$ & $(\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:13]{$13$}
} \\
\label{hilb39:108}
  $108$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:109}
  $109$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:108]{$108$}
} \\
\label{hilb39:110}
  $110$ & $(((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:109]{$109$}
} \\
\label{hilb39:111}
  $111$ & $((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  ((P\ \vee \  Q)\ \vee \  \neg \neg \neg (\neg \neg P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:106]{$106$}, \hyperref[hilb39:110]{$110$}} \\
\label{hilb39:112}
  $112$ & $(\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:107]{$107$}, \hyperref[hilb39:111]{$111$}} \\
\label{hilb39:113}
  $113$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)$ in \hyperref[hilb39:16]{$16$}
} \\
\label{hilb39:114}
  $114$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$ in \hyperref[hilb39:12]{$12$}
} \\
\label{hilb39:115}
  $115$ & $(D\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:114]{$114$}
} \\
\label{hilb39:116}
  $116$ & $((\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:115]{$115$}
} \\
\label{hilb39:117}
  $117$ & $((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:112]{$112$}, \hyperref[hilb39:116]{$116$}} \\
\label{hilb39:118}
  $118$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:113]{$113$}, \hyperref[hilb39:117]{$117$}} \\
\label{hilb39:119}
  $119$ & $(\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)$ in \hyperref[hilb39:19]{$19$}
} \\
\label{hilb39:120}
  $120$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q)$ in \hyperref[hilb39:114]{$114$}
} \\
\label{hilb39:121}
  $121$ & $((\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q)))\ \rightarrow \  (((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)$ in \hyperref[hilb39:120]{$120$}
} \\
\label{hilb39:122}
  $122$ & $((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \vee \  (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:119]{$119$}, \hyperref[hilb39:121]{$121$}} \\
\label{hilb39:123}
  $123$ & $(\neg \neg (\neg \neg P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:118]{$118$}, \hyperref[hilb39:122]{$122$}} \\
\label{hilb39:124}
  $124$ & $\neg \neg (\neg \neg P\ \vee \  \neg \neg Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb39:83]{$83$}, \hyperref[hilb39:123]{$123$}} \\
\label{hilb39:125}
  $125$ & $\neg (\neg P\ \wedge \  \neg Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb39:124]{$124$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



By duality we get the second distributive law (first direction):




\begin{thm}[hilb40]
\hypertarget{hilb40}{}
\begin{displaymath}
(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb40:1}
  $1$ & $((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb37}{hilb37} } \\
\label{hilb40:2}
  $2$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\
\label{hilb40:3}
  $3$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb40:2]{$2$}
} \\
\label{hilb40:4}
  $4$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb40:3]{$3$}
} \\
\label{hilb40:5}
  $5$ & $(B\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb40:4]{$4$}
} \\
\label{hilb40:6}
  $6$ & $(((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:5]{$5$}
} \\
\label{hilb40:7}
  $7$ & $\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:1]{$1$}, \hyperref[hilb40:6]{$6$}} \\
\label{hilb40:8}
  $8$ & $P\ \rightarrow \  \neg \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\
\label{hilb40:9}
  $9$ & $B\ \rightarrow \  \neg \neg B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb40:8]{$8$}
} \\
\label{hilb40:10}
  $10$ & $(Q\ \wedge \  A)\ \rightarrow \  \neg \neg (Q\ \wedge \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  A$ in \hyperref[hilb40:9]{$9$}
} \\
\label{hilb40:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb40:12}
  $12$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb40:11]{$11$}
} \\
\label{hilb40:13}
  $13$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb40:12]{$12$}
} \\
\label{hilb40:14}
  $14$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb40:13]{$13$}
} \\
\label{hilb40:15}
  $15$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:16}
  $16$ & $(D\ \rightarrow \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (Q\ \wedge \  A)$ in \hyperref[hilb40:15]{$15$}
} \\
\label{hilb40:17}
  $17$ & $((Q\ \wedge \  A)\ \rightarrow \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $Q\ \wedge \  A$ in \hyperref[hilb40:16]{$16$}
} \\
\label{hilb40:18}
  $18$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:10]{$10$}, \hyperref[hilb40:17]{$17$}} \\
\label{hilb40:19}
  $19$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:2]{$2$}
} \\
\label{hilb40:20}
  $20$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:19]{$19$}
} \\
\label{hilb40:21}
  $21$ & $(C\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  \neg \neg (Q\ \wedge \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:22}
  $22$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb40:21]{$21$}
} \\
\label{hilb40:23}
  $23$ & $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:18]{$18$}, \hyperref[hilb40:22]{$22$}} \\
\label{hilb40:24}
  $24$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\
\label{hilb40:25}
  $25$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\
\label{hilb40:26}
  $26$ & $(C\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:27}
  $27$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:26]{$26$}
} \\
\label{hilb40:28}
  $28$ & $\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:23]{$23$}, \hyperref[hilb40:27]{$27$}} \\
\label{hilb40:29}
  $29$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:30}
  $30$ & $(D\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:29]{$29$}
} \\
\label{hilb40:31}
  $31$ & $(\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:30]{$30$}
} \\
\label{hilb40:32}
  $32$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:28]{$28$}, \hyperref[hilb40:31]{$31$}} \\
\label{hilb40:33}
  $33$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{hilb40:34}
  $34$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:33]{$33$}
} \\
\label{hilb40:35}
  $35$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:34]{$34$}
} \\
\label{hilb40:36}
  $36$ & $(C\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:37}
  $37$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:36]{$36$}
} \\
\label{hilb40:38}
  $38$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb40:39}
  $39$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb40:38]{$38$}
} \\
\label{hilb40:40}
  $40$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb40:39]{$39$}
} \\
\label{hilb40:41}
  $41$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb40:40]{$40$}
} \\
\label{hilb40:42}
  $42$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:43}
  $43$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:42]{$42$}
} \\
\label{hilb40:44}
  $44$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:43]{$43$}
} \\
\label{hilb40:45}
  $45$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:37]{$37$}, \hyperref[hilb40:44]{$44$}} \\
\label{hilb40:46}
  $46$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:32]{$32$}, \hyperref[hilb40:45]{$45$}} \\
\label{hilb40:47}
  $47$ & $(C\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:48}
  $48$ & $(\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:47]{$47$}
} \\
\label{hilb40:49}
  $49$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:50}
  $50$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:49]{$49$}
} \\
\label{hilb40:51}
  $51$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:50]{$50$}
} \\
\label{hilb40:52}
  $52$ & $((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg \neg (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:46]{$46$}, \hyperref[hilb40:51]{$51$}} \\
\label{hilb40:53}
  $53$ & $(\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:48]{$48$}, \hyperref[hilb40:52]{$52$}} \\
\label{hilb40:54}
  $54$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:24]{$24$}
} \\
\label{hilb40:55}
  $55$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:54]{$54$}
} \\
\label{hilb40:56}
  $56$ & $(C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg C\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:55]{$55$}
} \\
\label{hilb40:57}
  $57$ & $(\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb40:56]{$56$}
} \\
\label{hilb40:58}
  $58$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:59}
  $59$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:58]{$58$}
} \\
\label{hilb40:60}
  $60$ & $((\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:59]{$59$}
} \\
\label{hilb40:61}
  $61$ & $((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:53]{$53$}, \hyperref[hilb40:60]{$60$}} \\
\label{hilb40:62}
  $62$ & $(\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:57]{$57$}, \hyperref[hilb40:61]{$61$}} \\
\label{hilb40:63}
  $63$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:25]{$25$}
} \\
\label{hilb40:64}
  $64$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:63]{$63$}
} \\
\label{hilb40:65}
  $65$ & $(\neg C\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:64]{$64$}
} \\
\label{hilb40:66}
  $66$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:65]{$65$}
} \\
\label{hilb40:67}
  $67$ & $(D\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:58]{$58$}
} \\
\label{hilb40:68}
  $68$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:67]{$67$}
} \\
\label{hilb40:69}
  $69$ & $((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:66]{$66$}, \hyperref[hilb40:68]{$68$}} \\
\label{hilb40:70}
  $70$ & $(\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:62]{$62$}, \hyperref[hilb40:69]{$69$}} \\
\label{hilb40:71}
  $71$ & $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:7]{$7$}, \hyperref[hilb40:70]{$70$}} \\
\label{hilb40:72}
  $72$ & $\neg \neg P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\
\label{hilb40:73}
  $73$ & $\neg \neg A\ \rightarrow \  A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb40:72]{$72$}
} \\
\label{hilb40:74}
  $74$ & $\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb40:73]{$73$}
} \\
\label{hilb40:75}
  $75$ & $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand1}{defand1} } \\
\label{hilb40:76}
  $76$ & $\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand2}{defand2} } \\
\label{hilb40:77}
  $77$ & $(B\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb40:4]{$4$}
} \\
\label{hilb40:78}
  $78$ & $(\neg \neg (P\ \vee \  Q)\ \rightarrow \  (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:77]{$77$}
} \\
\label{hilb40:79}
  $79$ & $\neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:74]{$74$}, \hyperref[hilb40:78]{$78$}} \\
\label{hilb40:80}
  $80$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:81}
  $81$ & $(D\ \rightarrow \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:80]{$80$}
} \\
\label{hilb40:82}
  $82$ & $(\neg (P\ \vee \  Q)\ \rightarrow \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb40:81]{$81$}
} \\
\label{hilb40:83}
  $83$ & $(\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:79]{$79$}, \hyperref[hilb40:82]{$82$}} \\
\label{hilb40:84}
  $84$ & $(C\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:85}
  $85$ & $(\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb40:84]{$84$}
} \\
\label{hilb40:86}
  $86$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:87}
  $87$ & $(D\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:86]{$86$}
} \\
\label{hilb40:88}
  $88$ & $((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:87]{$87$}
} \\
\label{hilb40:89}
  $89$ & $((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:85]{$85$}, \hyperref[hilb40:88]{$88$}} \\
\label{hilb40:90}
  $90$ & $(\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:83]{$83$}, \hyperref[hilb40:89]{$89$}} \\
\label{hilb40:91}
  $91$ & $(C\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:92}
  $92$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb40:91]{$91$}
} \\
\label{hilb40:93}
  $93$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:94}
  $94$ & $(D\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:93]{$93$}
} \\
\label{hilb40:95}
  $95$ & $((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb40:94]{$94$}
} \\
\label{hilb40:96}
  $96$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:90]{$90$}, \hyperref[hilb40:95]{$95$}} \\
\label{hilb40:97}
  $97$ & $(\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:92]{$92$}, \hyperref[hilb40:96]{$96$}} \\
\label{hilb40:98}
  $98$ & $(C\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:99}
  $99$ & $((\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:98]{$98$}
} \\
\label{hilb40:100}
  $100$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:97]{$97$}, \hyperref[hilb40:99]{$99$}} \\
\label{hilb40:101}
  $101$ & $(P\ \wedge \  B)\ \rightarrow \  \neg (\neg P\ \vee \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:75]{$75$}
} \\
\label{hilb40:102}
  $102$ & $(C\ \wedge \  B)\ \rightarrow \  \neg (\neg C\ \vee \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:101]{$101$}
} \\
\label{hilb40:103}
  $103$ & $(C\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg C\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb40:102]{$102$}
} \\
\label{hilb40:104}
  $104$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:103]{$103$}
} \\
\label{hilb40:105}
  $105$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:106}
  $106$ & $(D\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb40:105]{$105$}
} \\
\label{hilb40:107}
  $107$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb40:106]{$106$}
} \\
\label{hilb40:108}
  $108$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:100]{$100$}, \hyperref[hilb40:107]{$107$}} \\
\label{hilb40:109}
  $109$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:104]{$104$}, \hyperref[hilb40:108]{$108$}} \\
\label{hilb40:110}
  $110$ & $\neg (\neg P\ \vee \  \neg B)\ \rightarrow \  (P\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:76]{$76$}
} \\
\label{hilb40:111}
  $111$ & $\neg (\neg C\ \vee \  \neg B)\ \rightarrow \  (C\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:110]{$110$}
} \\
\label{hilb40:112}
  $112$ & $\neg (\neg C\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (C\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb40:111]{$111$}
} \\
\label{hilb40:113}
  $113$ & $\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb40:112]{$112$}
} \\
\label{hilb40:114}
  $114$ & $(D\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:105]{$105$}
} \\
\label{hilb40:115}
  $115$ & $(\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb40:114]{$114$}
} \\
\label{hilb40:116}
  $116$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:113]{$113$}, \hyperref[hilb40:115]{$115$}} \\
\label{hilb40:117}
  $117$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:109]{$109$}, \hyperref[hilb40:116]{$116$}} \\
\label{hilb40:118}
  $118$ & $(C\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:119}
  $119$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:118]{$118$}
} \\
\label{hilb40:120}
  $120$ & $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:117]{$117$}, \hyperref[hilb40:119]{$119$}} \\
\label{hilb40:121}
  $121$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:122}
  $122$ & $(D\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:121]{$121$}
} \\
\label{hilb40:123}
  $123$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:122]{$122$}
} \\
\label{hilb40:124}
  $124$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:120]{$120$}, \hyperref[hilb40:123]{$123$}} \\
\label{hilb40:125}
  $125$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:56]{$56$}
} \\
\label{hilb40:126}
  $126$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:127}
  $127$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:126]{$126$}
} \\
\label{hilb40:128}
  $128$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:127]{$127$}
} \\
\label{hilb40:129}
  $129$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:124]{$124$}, \hyperref[hilb40:128]{$128$}} \\
\label{hilb40:130}
  $130$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:125]{$125$}, \hyperref[hilb40:129]{$129$}} \\
\label{hilb40:131}
  $131$ & $(\neg C\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:64]{$64$}
} \\
\label{hilb40:132}
  $132$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:131]{$131$}
} \\
\label{hilb40:133}
  $133$ & $(D\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:126]{$126$}
} \\
\label{hilb40:134}
  $134$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:133]{$133$}
} \\
\label{hilb40:135}
  $135$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:132]{$132$}, \hyperref[hilb40:134]{$134$}} \\
\label{hilb40:136}
  $136$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:130]{$130$}, \hyperref[hilb40:135]{$135$}} \\
\label{hilb40:137}
  $137$ & $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:71]{$71$}, \hyperref[hilb40:136]{$136$}} \\
\label{hilb40:138}
  $138$ & $\neg \neg B\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb40:72]{$72$}
} \\
\label{hilb40:139}
  $139$ & $\neg \neg (P\ \vee \  A)\ \rightarrow \  (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb40:138]{$138$}
} \\
\label{hilb40:140}
  $140$ & $(C\ \rightarrow \  (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:141}
  $141$ & $(\neg \neg (P\ \vee \  A)\ \rightarrow \  (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \rightarrow \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:140]{$140$}
} \\
\label{hilb40:142}
  $142$ & $\neg (P\ \vee \  A)\ \rightarrow \  \neg \neg \neg (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:139]{$139$}, \hyperref[hilb40:141]{$141$}} \\
\label{hilb40:143}
  $143$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:144}
  $144$ & $(D\ \rightarrow \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:143]{$143$}
} \\
\label{hilb40:145}
  $145$ & $(\neg (P\ \vee \  A)\ \rightarrow \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb40:144]{$144$}
} \\
\label{hilb40:146}
  $146$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:142]{$142$}, \hyperref[hilb40:145]{$145$}} \\
\label{hilb40:147}
  $147$ & $(C\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:148}
  $148$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb40:147]{$147$}
} \\
\label{hilb40:149}
  $149$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:146]{$146$}, \hyperref[hilb40:148]{$148$}} \\
\label{hilb40:150}
  $150$ & $(C\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg C\ \vee \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:102]{$102$}
} \\
\label{hilb40:151}
  $151$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:150]{$150$}
} \\
\label{hilb40:152}
  $152$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:153}
  $153$ & $(D\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb40:152]{$152$}
} \\
\label{hilb40:154}
  $154$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:153]{$153$}
} \\
\label{hilb40:155}
  $155$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:149]{$149$}, \hyperref[hilb40:154]{$154$}} \\
\label{hilb40:156}
  $156$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:151]{$151$}, \hyperref[hilb40:155]{$155$}} \\
\label{hilb40:157}
  $157$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb40:112]{$112$}
} \\
\label{hilb40:158}
  $158$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:152]{$152$}
} \\
\label{hilb40:159}
  $159$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb40:158]{$158$}
} \\
\label{hilb40:160}
  $160$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:157]{$157$}, \hyperref[hilb40:159]{$159$}} \\
\label{hilb40:161}
  $161$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:156]{$156$}, \hyperref[hilb40:160]{$160$}} \\
\label{hilb40:162}
  $162$ & $(C\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:163}
  $163$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)$ in \hyperref[hilb40:162]{$162$}
} \\
\label{hilb40:164}
  $164$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:161]{$161$}, \hyperref[hilb40:163]{$163$}} \\
\label{hilb40:165}
  $165$ & $(D\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:121]{$121$}
} \\
\label{hilb40:166}
  $166$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:165]{$165$}
} \\
\label{hilb40:167}
  $167$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:164]{$164$}, \hyperref[hilb40:166]{$166$}} \\
\label{hilb40:168}
  $168$ & $(C\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg C\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:55]{$55$}
} \\
\label{hilb40:169}
  $169$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:168]{$168$}
} \\
\label{hilb40:170}
  $170$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:171}
  $171$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:170]{$170$}
} \\
\label{hilb40:172}
  $172$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb40:171]{$171$}
} \\
\label{hilb40:173}
  $173$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:167]{$167$}, \hyperref[hilb40:172]{$172$}} \\
\label{hilb40:174}
  $174$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:169]{$169$}, \hyperref[hilb40:173]{$173$}} \\
\label{hilb40:175}
  $175$ & $(\neg C\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:64]{$64$}
} \\
\label{hilb40:176}
  $176$ & $(\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb40:175]{$175$}
} \\
\label{hilb40:177}
  $177$ & $(D\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:170]{$170$}
} \\
\label{hilb40:178}
  $178$ & $((\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb40:177]{$177$}
} \\
\label{hilb40:179}
  $179$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:176]{$176$}, \hyperref[hilb40:178]{$178$}} \\
\label{hilb40:180}
  $180$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:174]{$174$}, \hyperref[hilb40:179]{$179$}} \\
\label{hilb40:181}
  $181$ & $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:137]{$137$}, \hyperref[hilb40:180]{$180$}} \\
\label{hilb40:182}
  $182$ & $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  B))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb40:181]{$181$}
} \\
\label{hilb40:183}
  $183$ & $\neg (P\ \vee \  \neg \neg (C\ \wedge \  B))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  C)\ \wedge \  \neg \neg (P\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb40:182]{$182$}
} \\
\label{hilb40:184}
  $184$ & $\neg (D\ \vee \  \neg \neg (C\ \wedge \  B))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  C)\ \wedge \  \neg \neg (D\ \vee \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb40:183]{$183$}
} \\
\label{hilb40:185}
  $185$ & $\neg (D\ \vee \  \neg \neg (C\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  C)\ \wedge \  \neg \neg (D\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg A$ in \hyperref[hilb40:184]{$184$}
} \\
\label{hilb40:186}
  $186$ & $\neg (D\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (D\ \vee \  \neg Q)\ \wedge \  \neg \neg (D\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb40:185]{$185$}
} \\
\label{hilb40:187}
  $187$ & $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb40:186]{$186$}
} \\
\label{hilb40:188}
  $188$ & $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:187]{$187$} at occurence $1$
} \\
\label{hilb40:189}
  $189$ & $(P\ \vee \  Q)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb38}{hilb38} } \\
\label{hilb40:190}
  $190$ & $(P\ \vee \  B)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:189]{$189$}
} \\
\label{hilb40:191}
  $191$ & $(C\ \vee \  B)\ \rightarrow \  \neg (\neg C\ \wedge \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:190]{$190$}
} \\
\label{hilb40:192}
  $192$ & $(C\ \vee \  A)\ \rightarrow \  \neg (\neg C\ \wedge \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb40:191]{$191$}
} \\
\label{hilb40:193}
  $193$ & $(Q\ \vee \  A)\ \rightarrow \  \neg (\neg Q\ \wedge \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q$ in \hyperref[hilb40:192]{$192$}
} \\
\label{hilb40:194}
  $194$ & $(C\ \rightarrow \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (\neg \neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:195}
  $195$ & $((Q\ \vee \  A)\ \rightarrow \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (\neg \neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q\ \vee \  A$ in \hyperref[hilb40:194]{$194$}
} \\
\label{hilb40:196}
  $196$ & $\neg \neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  \neg (Q\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:193]{$193$}, \hyperref[hilb40:195]{$195$}} \\
\label{hilb40:197}
  $197$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg P\ \vee \  D)\ \rightarrow \  (\neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:198}
  $198$ & $(D\ \rightarrow \  \neg (Q\ \vee \  A))\ \rightarrow \  ((\neg P\ \vee \  D)\ \rightarrow \  (\neg P\ \vee \  \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (Q\ \vee \  A)$ in \hyperref[hilb40:197]{$197$}
} \\
\label{hilb40:199}
  $199$ & $(\neg \neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  \neg (Q\ \vee \  A))\ \rightarrow \  ((\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (\neg P\ \vee \  \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:198]{$198$}
} \\
\label{hilb40:200}
  $200$ & $(\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (\neg P\ \vee \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:196]{$196$}, \hyperref[hilb40:199]{$199$}} \\
\label{hilb40:201}
  $201$ & $(C\ \rightarrow \  (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  \neg (Q\ \vee \  A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:202}
  $202$ & $((\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:201]{$201$}
} \\
\label{hilb40:203}
  $203$ & $\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:200]{$200$}, \hyperref[hilb40:202]{$202$}} \\
\label{hilb40:204}
  $204$ & $(C\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg C\ \vee \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  A$ in \hyperref[hilb40:102]{$102$}
} \\
\label{hilb40:205}
  $205$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb40:204]{$204$}
} \\
\label{hilb40:206}
  $206$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:207}
  $207$ & $(D\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:206]{$206$}
} \\
\label{hilb40:208}
  $208$ & $(\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$ in \hyperref[hilb40:207]{$207$}
} \\
\label{hilb40:209}
  $209$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:203]{$203$}, \hyperref[hilb40:208]{$208$}} \\
\label{hilb40:210}
  $210$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:205]{$205$}, \hyperref[hilb40:209]{$209$}} \\
\label{hilb40:211}
  $211$ & $\neg (\neg C\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (C\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:111]{$111$}
} \\
\label{hilb40:212}
  $212$ & $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb40:211]{$211$}
} \\
\label{hilb40:213}
  $213$ & $(D\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:206]{$206$}
} \\
\label{hilb40:214}
  $214$ & $(\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:213]{$213$}
} \\
\label{hilb40:215}
  $215$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:212]{$212$}, \hyperref[hilb40:214]{$214$}} \\
\label{hilb40:216}
  $216$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:210]{$210$}, \hyperref[hilb40:215]{$215$}} \\
\label{hilb40:217}
  $217$ & $(C\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:20]{$20$}
} \\
\label{hilb40:218}
  $218$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb40:217]{$217$}
} \\
\label{hilb40:219}
  $219$ & $\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (P\ \wedge \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:216]{$216$}, \hyperref[hilb40:218]{$218$}} \\
\label{hilb40:220}
  $220$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  D)\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:221}
  $221$ & $(D\ \rightarrow \  \neg (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  D)\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb40:220]{$220$}
} \\
\label{hilb40:222}
  $222$ & $(\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:221]{$221$}
} \\
\label{hilb40:223}
  $223$ & $(\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:219]{$219$}, \hyperref[hilb40:222]{$222$}} \\
\label{hilb40:224}
  $224$ & $(C\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:225}
  $225$ & $(\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:224]{$224$}
} \\
\label{hilb40:226}
  $226$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:227}
  $227$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:226]{$226$}
} \\
\label{hilb40:228}
  $228$ & $((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb40:227]{$227$}
} \\
\label{hilb40:229}
  $229$ & $((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  ((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:225]{$225$}, \hyperref[hilb40:228]{$228$}} \\
\label{hilb40:230}
  $230$ & $(\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:223]{$223$}, \hyperref[hilb40:229]{$229$}} \\
\label{hilb40:231}
  $231$ & $(C\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:35]{$35$}
} \\
\label{hilb40:232}
  $232$ & $(\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:231]{$231$}
} \\
\label{hilb40:233}
  $233$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:234}
  $234$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:233]{$233$}
} \\
\label{hilb40:235}
  $235$ & $((\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb40:234]{$234$}
} \\
\label{hilb40:236}
  $236$ & $((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \vee \  \neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:230]{$230$}, \hyperref[hilb40:235]{$235$}} \\
\label{hilb40:237}
  $237$ & $(\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:232]{$232$}, \hyperref[hilb40:236]{$236$}} \\
\label{hilb40:238}
  $238$ & $(C\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg C\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:55]{$55$}
} \\
\label{hilb40:239}
  $239$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb40:238]{$238$}
} \\
\label{hilb40:240}
  $240$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:241}
  $241$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:240]{$240$}
} \\
\label{hilb40:242}
  $242$ & $((\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:241]{$241$}
} \\
\label{hilb40:243}
  $243$ & $(((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:237]{$237$}, \hyperref[hilb40:242]{$242$}} \\
\label{hilb40:244}
  $244$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:239]{$239$}, \hyperref[hilb40:243]{$243$}} \\
\label{hilb40:245}
  $245$ & $(\neg C\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (C\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:64]{$64$}
} \\
\label{hilb40:246}
  $246$ & $(\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb40:245]{$245$}
} \\
\label{hilb40:247}
  $247$ & $(D\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:240]{$240$}
} \\
\label{hilb40:248}
  $248$ & $((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:247]{$247$}
} \\
\label{hilb40:249}
  $249$ & $(((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:246]{$246$}, \hyperref[hilb40:248]{$248$}} \\
\label{hilb40:250}
  $250$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:244]{$244$}, \hyperref[hilb40:249]{$249$}} \\
\label{hilb40:251}
  $251$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:188]{$188$}, \hyperref[hilb40:250]{$250$}} \\
\label{hilb40:252}
  $252$ & $\neg (\neg P\ \wedge \  \neg Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb39}{hilb39} } \\
\label{hilb40:253}
  $253$ & $\neg (\neg P\ \wedge \  \neg B)\ \rightarrow \  (P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb40:252]{$252$}
} \\
\label{hilb40:254}
  $254$ & $\neg (\neg C\ \wedge \  \neg B)\ \rightarrow \  (C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb40:253]{$253$}
} \\
\label{hilb40:255}
  $255$ & $\neg (\neg C\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \rightarrow \  (C\ \vee \  \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg A)$ in \hyperref[hilb40:254]{$254$}
} \\
\label{hilb40:256}
  $256$ & $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg P\ \vee \  \neg Q)$ in \hyperref[hilb40:255]{$255$}
} \\
\label{hilb40:257}
  $257$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb40:14]{$14$}
} \\
\label{hilb40:258}
  $258$ & $(D\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)$ in \hyperref[hilb40:257]{$257$}
} \\
\label{hilb40:259}
  $259$ & $(\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:258]{$258$}
} \\
\label{hilb40:260}
  $260$ & $(\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:256]{$256$}, \hyperref[hilb40:259]{$259$}} \\
\label{hilb40:261}
  $261$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb40:238]{$238$}
} \\
\label{hilb40:262}
  $262$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:41]{$41$}
} \\
\label{hilb40:263}
  $263$ & $(D\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:262]{$262$}
} \\
\label{hilb40:264}
  $264$ & $((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:263]{$263$}
} \\
\label{hilb40:265}
  $265$ & $(((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:260]{$260$}, \hyperref[hilb40:264]{$264$}} \\
\label{hilb40:266}
  $266$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:261]{$261$}, \hyperref[hilb40:265]{$265$}} \\
\label{hilb40:267}
  $267$ & $(\neg C\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (C\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)$ in \hyperref[hilb40:64]{$64$}
} \\
\label{hilb40:268}
  $268$ & $(\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb40:267]{$267$}
} \\
\label{hilb40:269}
  $269$ & $(D\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:262]{$262$}
} \\
\label{hilb40:270}
  $270$ & $((\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$ in \hyperref[hilb40:269]{$269$}
} \\
\label{hilb40:271}
  $271$ & $(((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  (\neg (P\ \wedge \  (Q\ \vee \  A))\ \vee \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))\ \rightarrow \  (((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:268]{$268$}, \hyperref[hilb40:270]{$270$}} \\
\label{hilb40:272}
  $272$ & $((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A)))\ \rightarrow \  ((P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:266]{$266$}, \hyperref[hilb40:271]{$271$}} \\
\label{hilb40:273}
  $273$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb40:251]{$251$}, \hyperref[hilb40:272]{$272$}} \\
\label{hilb40:274}
  $274$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \vee \  \neg (\neg P\ \vee \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:273]{$273$} at occurence $1$
} \\
\label{hilb40:275}
  $275$ & $(P\ \wedge \  (Q\ \vee \  A))\ \rightarrow \  ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb40:274]{$274$} at occurence $1$
} \\
 & & \qedhere
\end{longtable}
\end{proof}



The second distributive law (second direction):




\begin{thm}[hilb41]
\hypertarget{hilb41}{}
\begin{displaymath}
((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{hilb41:1}
  $1$ & $(P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb36}{hilb36} } \\
\label{hilb41:2}
  $2$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg Q\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb7}{hilb7} } \\
\label{hilb41:3}
  $3$ & $(P\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $A$ in \hyperref[hilb41:2]{$2$}
} \\
\label{hilb41:4}
  $4$ & $(B\ \rightarrow \  A)\ \rightarrow \  (\neg A\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb41:3]{$3$}
} \\
\label{hilb41:5}
  $5$ & $(B\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:4]{$4$}
} \\
\label{hilb41:6}
  $6$ & $((P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb41:5]{$5$}
} \\
\label{hilb41:7}
  $7$ & $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:1]{$1$}, \hyperref[hilb41:6]{$6$}} \\
\label{hilb41:8}
  $8$ & $\neg \neg P\ \rightarrow \  P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb6}{hilb6} } \\
\label{hilb41:9}
  $9$ & $\neg \neg B\ \rightarrow \  B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb41:8]{$8$}
} \\
\label{hilb41:10}
  $10$ & $\neg \neg (Q\ \wedge \  A)\ \rightarrow \  (Q\ \wedge \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \wedge \  A$ in \hyperref[hilb41:9]{$9$}
} \\
\label{hilb41:11}
  $11$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \vee \  P)\ \rightarrow \  (A\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom4}{axiom4} } \\
\label{hilb41:12}
  $12$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb41:11]{$11$}
} \\
\label{hilb41:13}
  $13$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  P)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb41:12]{$12$}
} \\
\label{hilb41:14}
  $14$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \vee \  D)\ \rightarrow \  (B\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb41:13]{$13$}
} \\
\label{hilb41:15}
  $15$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:16}
  $16$ & $(D\ \rightarrow \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  D)\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q\ \wedge \  A$ in \hyperref[hilb41:15]{$15$}
} \\
\label{hilb41:17}
  $17$ & $(\neg \neg (Q\ \wedge \  A)\ \rightarrow \  (Q\ \wedge \  A))\ \rightarrow \  ((P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (Q\ \wedge \  A)$ in \hyperref[hilb41:16]{$16$}
} \\
\label{hilb41:18}
  $18$ & $(P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:10]{$10$}, \hyperref[hilb41:17]{$17$}} \\
\label{hilb41:19}
  $19$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:2]{$2$}
} \\
\label{hilb41:20}
  $20$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg B\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:19]{$19$}
} \\
\label{hilb41:21}
  $21$ & $(C\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  (Q\ \wedge \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:22}
  $22$ & $((P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \rightarrow \  (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  \neg \neg (Q\ \wedge \  A)$ in \hyperref[hilb41:21]{$21$}
} \\
\label{hilb41:23}
  $23$ & $\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:18]{$18$}, \hyperref[hilb41:22]{$22$}} \\
\label{hilb41:24}
  $24$ & $(P\ \rightarrow \  Q)\ \rightarrow \  (\neg P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl1}{defimpl1} } \\
\label{hilb41:25}
  $25$ & $(\neg P\ \vee \  Q)\ \rightarrow \  (P\ \rightarrow \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defimpl2}{defimpl2} } \\
\label{hilb41:26}
  $26$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:27}
  $27$ & $(D\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  D)\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:26]{$26$}
} \\
\label{hilb41:28}
  $28$ & $(\neg (P\ \vee \  (Q\ \wedge \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb41:27]{$27$}
} \\
\label{hilb41:29}
  $29$ & $(\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:23]{$23$}, \hyperref[hilb41:28]{$28$}} \\
\label{hilb41:30}
  $30$ & $(P\ \rightarrow \  B)\ \rightarrow \  (\neg P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:24]{$24$}
} \\
\label{hilb41:31}
  $31$ & $(C\ \rightarrow \  B)\ \rightarrow \  (\neg C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:30]{$30$}
} \\
\label{hilb41:32}
  $32$ & $(C\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg C\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb41:31]{$31$}
} \\
\label{hilb41:33}
  $33$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:32]{$32$}
} \\
\label{hilb41:34}
  $34$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((A\ \rightarrow \  P)\ \rightarrow \  (A\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb1}{hilb1} } \\
\label{hilb41:35}
  $35$ & $(P\ \rightarrow \  Q)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb41:34]{$34$}
} \\
\label{hilb41:36}
  $36$ & $(P\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  P)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb41:35]{$35$}
} \\
\label{hilb41:37}
  $37$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((B\ \rightarrow \  D)\ \rightarrow \  (B\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb41:36]{$36$}
} \\
\label{hilb41:38}
  $38$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:39}
  $39$ & $(D\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:38]{$38$}
} \\
\label{hilb41:40}
  $40$ & $((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A))$ in \hyperref[hilb41:39]{$39$}
} \\
\label{hilb41:41}
  $41$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:29]{$29$}, \hyperref[hilb41:40]{$40$}} \\
\label{hilb41:42}
  $42$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:33]{$33$}, \hyperref[hilb41:41]{$41$}} \\
\label{hilb41:43}
  $43$ & $(\neg P\ \vee \  B)\ \rightarrow \  (P\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:25]{$25$}
} \\
\label{hilb41:44}
  $44$ & $(\neg C\ \vee \  B)\ \rightarrow \  (C\ \rightarrow \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:43]{$43$}
} \\
\label{hilb41:45}
  $45$ & $(\neg C\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (C\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:44]{$44$}
} \\
\label{hilb41:46}
  $46$ & $(\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:45]{$45$}
} \\
\label{hilb41:47}
  $47$ & $(D\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:38]{$38$}
} \\
\label{hilb41:48}
  $48$ & $((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:47]{$47$}
} \\
\label{hilb41:49}
  $49$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:46]{$46$}, \hyperref[hilb41:48]{$48$}} \\
\label{hilb41:50}
  $50$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  (Q\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:42]{$42$}, \hyperref[hilb41:49]{$49$}} \\
\label{hilb41:51}
  $51$ & $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:7]{$7$}, \hyperref[hilb41:50]{$50$}} \\
\label{hilb41:52}
  $52$ & $P\ \rightarrow \  \neg \neg P$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{hilb5}{hilb5} } \\
\label{hilb41:53}
  $53$ & $A\ \rightarrow \  \neg \neg A$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $A$ in \hyperref[hilb41:52]{$52$}
} \\
\label{hilb41:54}
  $54$ & $(P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $P\ \vee \  Q$ in \hyperref[hilb41:53]{$53$}
} \\
\label{hilb41:55}
  $55$ & $(P\ \wedge \  Q)\ \rightarrow \  \neg (\neg P\ \vee \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand1}{defand1} } \\
\label{hilb41:56}
  $56$ & $\neg (\neg P\ \vee \  \neg Q)\ \rightarrow \  (P\ \wedge \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{prophilbert1_1.00.00_1.00.00.pdf}{}{defand2}{defand2} } \\
\label{hilb41:57}
  $57$ & $(B\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:4]{$4$}
} \\
\label{hilb41:58}
  $58$ & $((P\ \vee \  Q)\ \rightarrow \  \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  Q$ in \hyperref[hilb41:57]{$57$}
} \\
\label{hilb41:59}
  $59$ & $\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:54]{$54$}, \hyperref[hilb41:58]{$58$}} \\
\label{hilb41:60}
  $60$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:61}
  $61$ & $(D\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb41:60]{$60$}
} \\
\label{hilb41:62}
  $62$ & $(\neg \neg \neg (P\ \vee \  Q)\ \rightarrow \  \neg (P\ \vee \  Q))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:61]{$61$}
} \\
\label{hilb41:63}
  $63$ & $(\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:59]{$59$}, \hyperref[hilb41:62]{$62$}} \\
\label{hilb41:64}
  $64$ & $(P\ \vee \  Q)\ \rightarrow \  (Q\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{axiom3}{axiom3} } \\
\label{hilb41:65}
  $65$ & $(P\ \vee \  B)\ \rightarrow \  (B\ \vee \  P)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:64]{$64$}
} \\
\label{hilb41:66}
  $66$ & $(C\ \vee \  B)\ \rightarrow \  (B\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:65]{$65$}
} \\
\label{hilb41:67}
  $67$ & $(C\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:68}
  $68$ & $(\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb41:67]{$67$}
} \\
\label{hilb41:69}
  $69$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:70}
  $70$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:69]{$69$}
} \\
\label{hilb41:71}
  $71$ & $((\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)$ in \hyperref[hilb41:70]{$70$}
} \\
\label{hilb41:72}
  $72$ & $((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:68]{$68$}, \hyperref[hilb41:71]{$71$}} \\
\label{hilb41:73}
  $73$ & $(\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:63]{$63$}, \hyperref[hilb41:72]{$72$}} \\
\label{hilb41:74}
  $74$ & $(C\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:75}
  $75$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:74]{$74$}
} \\
\label{hilb41:76}
  $76$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:77}
  $77$ & $(D\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:76]{$76$}
} \\
\label{hilb41:78}
  $78$ & $((\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:77]{$77$}
} \\
\label{hilb41:79}
  $79$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  A)\ \vee \  \neg \neg \neg (P\ \vee \  Q)))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:73]{$73$}, \hyperref[hilb41:78]{$78$}} \\
\label{hilb41:80}
  $80$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:75]{$75$}, \hyperref[hilb41:79]{$79$}} \\
\label{hilb41:81}
  $81$ & $(C\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:82}
  $82$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:81]{$81$}
} \\
\label{hilb41:83}
  $83$ & $\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:80]{$80$}, \hyperref[hilb41:82]{$82$}} \\
\label{hilb41:84}
  $84$ & $(P\ \wedge \  B)\ \rightarrow \  \neg (\neg P\ \vee \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:55]{$55$}
} \\
\label{hilb41:85}
  $85$ & $(C\ \wedge \  B)\ \rightarrow \  \neg (\neg C\ \vee \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:84]{$84$}
} \\
\label{hilb41:86}
  $86$ & $(C\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg C\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb41:85]{$85$}
} \\
\label{hilb41:87}
  $87$ & $((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  Q$ in \hyperref[hilb41:86]{$86$}
} \\
\label{hilb41:88}
  $88$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:89}
  $89$ & $(D\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb41:88]{$88$}
} \\
\label{hilb41:90}
  $90$ & $(\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb41:89]{$89$}
} \\
\label{hilb41:91}
  $91$ & $(((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:83]{$83$}, \hyperref[hilb41:90]{$90$}} \\
\label{hilb41:92}
  $92$ & $((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:87]{$87$}, \hyperref[hilb41:91]{$91$}} \\
\label{hilb41:93}
  $93$ & $\neg (\neg P\ \vee \  \neg B)\ \rightarrow \  (P\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:56]{$56$}
} \\
\label{hilb41:94}
  $94$ & $\neg (\neg C\ \vee \  \neg B)\ \rightarrow \  (C\ \wedge \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:93]{$93$}
} \\
\label{hilb41:95}
  $95$ & $\neg (\neg C\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (C\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb41:94]{$94$}
} \\
\label{hilb41:96}
  $96$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:95]{$95$}
} \\
\label{hilb41:97}
  $97$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:88]{$88$}
} \\
\label{hilb41:98}
  $98$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb41:97]{$97$}
} \\
\label{hilb41:99}
  $99$ & $(((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:96]{$96$}, \hyperref[hilb41:98]{$98$}} \\
\label{hilb41:100}
  $100$ & $((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:92]{$92$}, \hyperref[hilb41:99]{$99$}} \\
\label{hilb41:101}
  $101$ & $(C\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:102}
  $102$ & $(((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:101]{$101$}
} \\
\label{hilb41:103}
  $103$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:100]{$100$}, \hyperref[hilb41:102]{$102$}} \\
\label{hilb41:104}
  $104$ & $(C\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:105}
  $105$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:104]{$104$}
} \\
\label{hilb41:106}
  $106$ & $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:103]{$103$}, \hyperref[hilb41:105]{$105$}} \\
\label{hilb41:107}
  $107$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:108}
  $108$ & $(D\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:107]{$107$}
} \\
\label{hilb41:109}
  $109$ & $(\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:108]{$108$}
} \\
\label{hilb41:110}
  $110$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:106]{$106$}, \hyperref[hilb41:109]{$109$}} \\
\label{hilb41:111}
  $111$ & $(C\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:112}
  $112$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:111]{$111$}
} \\
\label{hilb41:113}
  $113$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:114}
  $114$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:113]{$113$}
} \\
\label{hilb41:115}
  $115$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:114]{$114$}
} \\
\label{hilb41:116}
  $116$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:112]{$112$}, \hyperref[hilb41:115]{$115$}} \\
\label{hilb41:117}
  $117$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:110]{$110$}, \hyperref[hilb41:116]{$116$}} \\
\label{hilb41:118}
  $118$ & $(C\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:119}
  $119$ & $(\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:118]{$118$}
} \\
\label{hilb41:120}
  $120$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:121}
  $121$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:120]{$120$}
} \\
\label{hilb41:122}
  $122$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:121]{$121$}
} \\
\label{hilb41:123}
  $123$ & $((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:117]{$117$}, \hyperref[hilb41:122]{$122$}} \\
\label{hilb41:124}
  $124$ & $(\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:119]{$119$}, \hyperref[hilb41:123]{$123$}} \\
\label{hilb41:125}
  $125$ & $(C\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg C\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:31]{$31$}
} \\
\label{hilb41:126}
  $126$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:125]{$125$}
} \\
\label{hilb41:127}
  $127$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:128}
  $128$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:127]{$127$}
} \\
\label{hilb41:129}
  $129$ & $((\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:128]{$128$}
} \\
\label{hilb41:130}
  $130$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:124]{$124$}, \hyperref[hilb41:129]{$129$}} \\
\label{hilb41:131}
  $131$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:126]{$126$}, \hyperref[hilb41:130]{$130$}} \\
\label{hilb41:132}
  $132$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:45]{$45$}
} \\
\label{hilb41:133}
  $133$ & $(D\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:127]{$127$}
} \\
\label{hilb41:134}
  $134$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:133]{$133$}
} \\
\label{hilb41:135}
  $135$ & $((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:132]{$132$}, \hyperref[hilb41:134]{$134$}} \\
\label{hilb41:136}
  $136$ & $(\neg ((P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:131]{$131$}, \hyperref[hilb41:135]{$135$}} \\
\label{hilb41:137}
  $137$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:51]{$51$}, \hyperref[hilb41:136]{$136$}} \\
\label{hilb41:138}
  $138$ & $B\ \rightarrow \  \neg \neg B$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $B$ in \hyperref[hilb41:52]{$52$}
} \\
\label{hilb41:139}
  $139$ & $(P\ \vee \  A)\ \rightarrow \  \neg \neg (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \vee \  A$ in \hyperref[hilb41:138]{$138$}
} \\
\label{hilb41:140}
  $140$ & $(C\ \rightarrow \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:141}
  $141$ & $((P\ \vee \  A)\ \rightarrow \  \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \vee \  A$ in \hyperref[hilb41:140]{$140$}
} \\
\label{hilb41:142}
  $142$ & $\neg \neg \neg (P\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:139]{$139$}, \hyperref[hilb41:141]{$141$}} \\
\label{hilb41:143}
  $143$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:144}
  $144$ & $(D\ \rightarrow \  \neg (P\ \vee \  A))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  D)\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  A)$ in \hyperref[hilb41:143]{$143$}
} \\
\label{hilb41:145}
  $145$ & $(\neg \neg \neg (P\ \vee \  A)\ \rightarrow \  \neg (P\ \vee \  A))\ \rightarrow \  ((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:144]{$144$}
} \\
\label{hilb41:146}
  $146$ & $(\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:142]{$142$}, \hyperref[hilb41:145]{$145$}} \\
\label{hilb41:147}
  $147$ & $(C\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:148}
  $148$ & $((\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:147]{$147$}
} \\
\label{hilb41:149}
  $149$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:146]{$146$}, \hyperref[hilb41:148]{$148$}} \\
\label{hilb41:150}
  $150$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:86]{$86$}
} \\
\label{hilb41:151}
  $151$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:152}
  $152$ & $(D\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:151]{$151$}
} \\
\label{hilb41:153}
  $153$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A))$ in \hyperref[hilb41:152]{$152$}
} \\
\label{hilb41:154}
  $154$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:149]{$149$}, \hyperref[hilb41:153]{$153$}} \\
\label{hilb41:155}
  $155$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:150]{$150$}, \hyperref[hilb41:154]{$154$}} \\
\label{hilb41:156}
  $156$ & $\neg (\neg C\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (C\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:94]{$94$}
} \\
\label{hilb41:157}
  $157$ & $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)$ in \hyperref[hilb41:156]{$156$}
} \\
\label{hilb41:158}
  $158$ & $(D\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:151]{$151$}
} \\
\label{hilb41:159}
  $159$ & $(\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:158]{$158$}
} \\
\label{hilb41:160}
  $160$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg \neg (P\ \vee \  Q)\ \vee \  \neg \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:157]{$157$}, \hyperref[hilb41:159]{$159$}} \\
\label{hilb41:161}
  $161$ & $(\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:155]{$155$}, \hyperref[hilb41:160]{$160$}} \\
\label{hilb41:162}
  $162$ & $(C\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:163}
  $163$ & $((\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)$ in \hyperref[hilb41:162]{$162$}
} \\
\label{hilb41:164}
  $164$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:161]{$161$}, \hyperref[hilb41:163]{$163$}} \\
\label{hilb41:165}
  $165$ & $(C\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:166}
  $166$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:165]{$165$}
} \\
\label{hilb41:167}
  $167$ & $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:164]{$164$}, \hyperref[hilb41:166]{$166$}} \\
\label{hilb41:168}
  $168$ & $(D\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:107]{$107$}
} \\
\label{hilb41:169}
  $169$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:168]{$168$}
} \\
\label{hilb41:170}
  $170$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:167]{$167$}, \hyperref[hilb41:169]{$169$}} \\
\label{hilb41:171}
  $171$ & $(C\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:172}
  $172$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:171]{$171$}
} \\
\label{hilb41:173}
  $173$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:174}
  $174$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:173]{$173$}
} \\
\label{hilb41:175}
  $175$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:174]{$174$}
} \\
\label{hilb41:176}
  $176$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))))\ \rightarrow \  ((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:172]{$172$}, \hyperref[hilb41:175]{$175$}} \\
\label{hilb41:177}
  $177$ & $(\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:170]{$170$}, \hyperref[hilb41:176]{$176$}} \\
\label{hilb41:178}
  $178$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:118]{$118$}
} \\
\label{hilb41:179}
  $179$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:180}
  $180$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:179]{$179$}
} \\
\label{hilb41:181}
  $181$ & $((\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:180]{$180$}
} \\
\label{hilb41:182}
  $182$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))\ \vee \  \neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))))\ \rightarrow \  ((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:177]{$177$}, \hyperref[hilb41:181]{$181$}} \\
\label{hilb41:183}
  $183$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:178]{$178$}, \hyperref[hilb41:182]{$182$}} \\
\label{hilb41:184}
  $184$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))$ in \hyperref[hilb41:125]{$125$}
} \\
\label{hilb41:185}
  $185$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:186}
  $186$ & $(D\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:185]{$185$}
} \\
\label{hilb41:187}
  $187$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:186]{$186$}
} \\
\label{hilb41:188}
  $188$ & $((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:183]{$183$}, \hyperref[hilb41:187]{$187$}} \\
\label{hilb41:189}
  $189$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:184]{$184$}, \hyperref[hilb41:188]{$188$}} \\
\label{hilb41:190}
  $190$ & $(\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))$ in \hyperref[hilb41:45]{$45$}
} \\
\label{hilb41:191}
  $191$ & $(D\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:185]{$185$}
} \\
\label{hilb41:192}
  $192$ & $((\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  (((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$ in \hyperref[hilb41:191]{$191$}
} \\
\label{hilb41:193}
  $193$ & $((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \vee \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))\ \rightarrow \  ((\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:190]{$190$}, \hyperref[hilb41:192]{$192$}} \\
\label{hilb41:194}
  $194$ & $(\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))\ \rightarrow \  (\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:189]{$189$}, \hyperref[hilb41:193]{$193$}} \\
\label{hilb41:195}
  $195$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  A))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:137]{$137$}, \hyperref[hilb41:194]{$194$}} \\
\label{hilb41:196}
  $196$ & $\neg (\neg \neg (P\ \vee \  Q)\ \wedge \  \neg \neg (P\ \vee \  B))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (Q\ \wedge \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $A$ by $B$ in \hyperref[hilb41:195]{$195$}
} \\
\label{hilb41:197}
  $197$ & $\neg (\neg \neg (P\ \vee \  C)\ \wedge \  \neg \neg (P\ \vee \  B))\ \rightarrow \  \neg (P\ \vee \  \neg \neg (C\ \wedge \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $C$ in \hyperref[hilb41:196]{$196$}
} \\
\label{hilb41:198}
  $198$ & $\neg (\neg \neg (D\ \vee \  C)\ \wedge \  \neg \neg (D\ \vee \  B))\ \rightarrow \  \neg (D\ \vee \  \neg \neg (C\ \wedge \  B))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $D$ in \hyperref[hilb41:197]{$197$}
} \\
\label{hilb41:199}
  $199$ & $\neg (\neg \neg (D\ \vee \  C)\ \wedge \  \neg \neg (D\ \vee \  \neg A))\ \rightarrow \  \neg (D\ \vee \  \neg \neg (C\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg A$ in \hyperref[hilb41:198]{$198$}
} \\
\label{hilb41:200}
  $200$ & $\neg (\neg \neg (D\ \vee \  \neg Q)\ \wedge \  \neg \neg (D\ \vee \  \neg A))\ \rightarrow \  \neg (D\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg Q$ in \hyperref[hilb41:199]{$199$}
} \\
\label{hilb41:201}
  $201$ & $\neg (\neg \neg (\neg P\ \vee \  \neg Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg P$ in \hyperref[hilb41:200]{$200$}
} \\
\label{hilb41:202}
  $202$ & $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg \neg (\neg P\ \vee \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:201]{$201$} at occurence $1$
} \\
\label{hilb41:203}
  $203$ & $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:202]{$202$} at occurence $1$
} \\
\label{hilb41:204}
  $204$ & $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{and}{and} in \hyperref[hilb41:203]{$203$} at occurence $1$
} \\
\label{hilb41:205}
  $205$ & $(P\ \vee \  Q)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb38}{hilb38} } \\
\label{hilb41:206}
  $206$ & $(P\ \vee \  B)\ \rightarrow \  \neg (\neg P\ \wedge \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:205]{$205$}
} \\
\label{hilb41:207}
  $207$ & $(C\ \vee \  B)\ \rightarrow \  \neg (\neg C\ \wedge \  \neg B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:206]{$206$}
} \\
\label{hilb41:208}
  $208$ & $(C\ \vee \  (P\ \wedge \  A))\ \rightarrow \  \neg (\neg C\ \wedge \  \neg (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  A$ in \hyperref[hilb41:207]{$207$}
} \\
\label{hilb41:209}
  $209$ & $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  Q$ in \hyperref[hilb41:208]{$208$}
} \\
\label{hilb41:210}
  $210$ & $(C\ \rightarrow \  \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:211}
  $211$ & $(((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)$ in \hyperref[hilb41:210]{$210$}
} \\
\label{hilb41:212}
  $212$ & $\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:209]{$209$}, \hyperref[hilb41:211]{$211$}} \\
\label{hilb41:213}
  $213$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  D)\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:214}
  $214$ & $(D\ \rightarrow \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  D)\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$ in \hyperref[hilb41:213]{$213$}
} \\
\label{hilb41:215}
  $215$ & $(\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:214]{$214$}
} \\
\label{hilb41:216}
  $216$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:212]{$212$}, \hyperref[hilb41:215]{$215$}} \\
\label{hilb41:217}
  $217$ & $(C\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:218}
  $218$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:217]{$217$}
} \\
\label{hilb41:219}
  $219$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:220}
  $220$ & $(D\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  D)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:219]{$219$}
} \\
\label{hilb41:221}
  $221$ & $(((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$ in \hyperref[hilb41:220]{$220$}
} \\
\label{hilb41:222}
  $222$ & $(((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:218]{$218$}, \hyperref[hilb41:221]{$221$}} \\
\label{hilb41:223}
  $223$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:216]{$216$}, \hyperref[hilb41:222]{$222$}} \\
\label{hilb41:224}
  $224$ & $(C\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:66]{$66$}
} \\
\label{hilb41:225}
  $225$ & $(\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:224]{$224$}
} \\
\label{hilb41:226}
  $226$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:227}
  $227$ & $(D\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:226]{$226$}
} \\
\label{hilb41:228}
  $228$ & $(((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))))\ \rightarrow \  ((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:227]{$227$}
} \\
\label{hilb41:229}
  $229$ & $((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \vee \  \neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))))\ \rightarrow \  ((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:223]{$223$}, \hyperref[hilb41:228]{$228$}} \\
\label{hilb41:230}
  $230$ & $(\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:225]{$225$}, \hyperref[hilb41:229]{$229$}} \\
\label{hilb41:231}
  $231$ & $(C\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg C\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:31]{$31$}
} \\
\label{hilb41:232}
  $232$ & $(\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))$ in \hyperref[hilb41:231]{$231$}
} \\
\label{hilb41:233}
  $233$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:234}
  $234$ & $(D\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:233]{$233$}
} \\
\label{hilb41:235}
  $235$ & $((\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:234]{$234$}
} \\
\label{hilb41:236}
  $236$ & $((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg \neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:230]{$230$}, \hyperref[hilb41:235]{$235$}} \\
\label{hilb41:237}
  $237$ & $(\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:232]{$232$}, \hyperref[hilb41:236]{$236$}} \\
\label{hilb41:238}
  $238$ & $(\neg C\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (C\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:44]{$44$}
} \\
\label{hilb41:239}
  $239$ & $(\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)$ in \hyperref[hilb41:238]{$238$}
} \\
\label{hilb41:240}
  $240$ & $(D\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:233]{$233$}
} \\
\label{hilb41:241}
  $241$ & $((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  (((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:240]{$240$}
} \\
\label{hilb41:242}
  $242$ & $((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:239]{$239$}, \hyperref[hilb41:241]{$241$}} \\
\label{hilb41:243}
  $243$ & $(\neg (\neg (P\ \wedge \  Q)\ \wedge \  \neg (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:237]{$237$}, \hyperref[hilb41:242]{$242$}} \\
\label{hilb41:244}
  $244$ & $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:204]{$204$}, \hyperref[hilb41:243]{$243$}} \\
\label{hilb41:245}
  $245$ & $\neg (\neg P\ \wedge \  \neg Q)\ \rightarrow \  (P\ \vee \  Q)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule3}{add sentence} \hyperref{}{}{hilb39}{hilb39} } \\
\label{hilb41:246}
  $246$ & $\neg (\neg P\ \wedge \  \neg B)\ \rightarrow \  (P\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $B$ in \hyperref[hilb41:245]{$245$}
} \\
\label{hilb41:247}
  $247$ & $\neg (\neg C\ \wedge \  \neg B)\ \rightarrow \  (C\ \vee \  B)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $C$ in \hyperref[hilb41:246]{$246$}
} \\
\label{hilb41:248}
  $248$ & $\neg (\neg C\ \wedge \  \neg A)\ \rightarrow \  (C\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $A$ in \hyperref[hilb41:247]{$247$}
} \\
\label{hilb41:249}
  $249$ & $\neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  (Q\ \vee \  A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $Q$ in \hyperref[hilb41:248]{$248$}
} \\
\label{hilb41:250}
  $250$ & $(C\ \rightarrow \  (Q\ \vee \  A))\ \rightarrow \  (\neg (Q\ \vee \  A)\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  A$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:251}
  $251$ & $(\neg (\neg Q\ \wedge \  \neg A)\ \rightarrow \  (Q\ \vee \  A))\ \rightarrow \  (\neg (Q\ \vee \  A)\ \rightarrow \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:250]{$250$}
} \\
\label{hilb41:252}
  $252$ & $\neg (Q\ \vee \  A)\ \rightarrow \  \neg \neg (\neg Q\ \wedge \  \neg A)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:249]{$249$}, \hyperref[hilb41:251]{$251$}} \\
\label{hilb41:253}
  $253$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg P\ \vee \  D)\ \rightarrow \  (\neg P\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:254}
  $254$ & $(D\ \rightarrow \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  ((\neg P\ \vee \  D)\ \rightarrow \  (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:253]{$253$}
} \\
\label{hilb41:255}
  $255$ & $(\neg (Q\ \vee \  A)\ \rightarrow \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  ((\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (Q\ \vee \  A)$ in \hyperref[hilb41:254]{$254$}
} \\
\label{hilb41:256}
  $256$ & $(\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:252]{$252$}, \hyperref[hilb41:255]{$255$}} \\
\label{hilb41:257}
  $257$ & $(C\ \rightarrow \  (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg C)$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:20]{$20$}
} \\
\label{hilb41:258}
  $258$ & $((\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg P\ \vee \  \neg (Q\ \vee \  A)$ in \hyperref[hilb41:257]{$257$}
} \\
\label{hilb41:259}
  $259$ & $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:256]{$256$}, \hyperref[hilb41:258]{$258$}} \\
\label{hilb41:260}
  $260$ & $(C\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg C\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:85]{$85$}
} \\
\label{hilb41:261}
  $261$ & $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb41:260]{$260$}
} \\
\label{hilb41:262}
  $262$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:263}
  $263$ & $(D\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$ in \hyperref[hilb41:262]{$262$}
} \\
\label{hilb41:264}
  $264$ & $(\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:263]{$263$}
} \\
\label{hilb41:265}
  $265$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:259]{$259$}, \hyperref[hilb41:264]{$264$}} \\
\label{hilb41:266}
  $266$ & $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:261]{$261$}, \hyperref[hilb41:265]{$265$}} \\
\label{hilb41:267}
  $267$ & $\neg (\neg C\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (C\ \wedge \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $Q\ \vee \  A$ in \hyperref[hilb41:94]{$94$}
} \\
\label{hilb41:268}
  $268$ & $\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P$ in \hyperref[hilb41:267]{$267$}
} \\
\label{hilb41:269}
  $269$ & $(D\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  D)\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb41:262]{$262$}
} \\
\label{hilb41:270}
  $270$ & $(\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg (\neg P\ \vee \  \neg (Q\ \vee \  A))$ in \hyperref[hilb41:269]{$269$}
} \\
\label{hilb41:271}
  $271$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  \neg (\neg P\ \vee \  \neg (Q\ \vee \  A)))\ \rightarrow \  ((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:268]{$268$}, \hyperref[hilb41:270]{$270$}} \\
\label{hilb41:272}
  $272$ & $(P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:266]{$266$}, \hyperref[hilb41:271]{$271$}} \\
\label{hilb41:273}
  $273$ & $(D\ \rightarrow \  C)\ \rightarrow \  ((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))$ in \hyperref[hilb41:14]{$14$}
} \\
\label{hilb41:274}
  $274$ & $(D\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  ((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  D)\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb41:273]{$273$}
} \\
\label{hilb41:275}
  $275$ & $((P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  ((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)$ in \hyperref[hilb41:274]{$274$}
} \\
\label{hilb41:276}
  $276$ & $(\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:272]{$272$}, \hyperref[hilb41:275]{$275$}} \\
\label{hilb41:277}
  $277$ & $(((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)$ in \hyperref[hilb41:231]{$231$}
} \\
\label{hilb41:278}
  $278$ & $(D\ \rightarrow \  C)\ \rightarrow \  (((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  C))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:37]{$37$}
} \\
\label{hilb41:279}
  $279$ & $(D\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  (((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb41:278]{$278$}
} \\
\label{hilb41:280}
  $280$ & $((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  (((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))$ in \hyperref[hilb41:279]{$279$}
} \\
\label{hilb41:281}
  $281$ & $((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A))))\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:276]{$276$}, \hyperref[hilb41:280]{$280$}} \\
\label{hilb41:282}
  $282$ & $(((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:277]{$277$}, \hyperref[hilb41:281]{$281$}} \\
\label{hilb41:283}
  $283$ & $(\neg C\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (C\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $B$ by $P\ \wedge \  (Q\ \vee \  A)$ in \hyperref[hilb41:44]{$44$}
} \\
\label{hilb41:284}
  $284$ & $(\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $(P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A)$ in \hyperref[hilb41:283]{$283$}
} \\
\label{hilb41:285}
  $285$ & $(D\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  (((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  D)\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $C$ by $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb41:278]{$278$}
} \\
\label{hilb41:286}
  $286$ & $((\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  (((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $D$ by $\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))$ in \hyperref[hilb41:285]{$285$}
} \\
\label{hilb41:287}
  $287$ & $((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (\neg ((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \vee \  (P\ \wedge \  (Q\ \vee \  A))))\ \rightarrow \  ((((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:284]{$284$}, \hyperref[hilb41:286]{$286$}} \\
\label{hilb41:288}
  $288$ & $(((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  \neg (\neg Q\ \wedge \  \neg A)))\ \rightarrow \  (((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:282]{$282$}, \hyperref[hilb41:287]{$287$}} \\
\label{hilb41:289}
  $289$ & $((P\ \wedge \  Q)\ \vee \  (P\ \wedge \  A))\ \rightarrow \  (P\ \wedge \  (Q\ \vee \  A))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[hilb41:244]{$244$}, \hyperref[hilb41:288]{$288$}} \\
 & & \qedhere
\end{longtable}
\end{proof}


\section{Cross Reference}

This module is used by the following modules:

\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & predtheo2 \\
Version: & 1.00.00 \\
Rule version: & 1.02.00 \\
Orgin: & \url{predtheo2_1.00.00_1.02.00.qedeq}  \\
pdf: & \url{predtheo2_1.00.00_1.02.00.pdf}  \\
\end{longtable}


\end{document}
