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\begin{document}

\title{First theorems of Predicate Calculus}
\author{Michael Meyling}
\email{principa@meyling.com}
\date{2002-07-27T21:00:49}

\begin{abstract}
This module includes first proofs of predicate calculus theorems.
\end{abstract}

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\section*{module specification}

Author of this module: 
\begin{longtable}[h!]{l@{\extracolsep{\fill}}l}
Michael Meyling & michael@meyling.com \\
\end{longtable}


\medskip
This module has the following specification:

\mbox{}
\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & predtheo1 \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{http://www.meyling.com/principia/0_00_51/predtheo1_1.00.00_1.00.00.qedeq} \\
\end{longtable}


\medskip
The following modules were used:

\mbox{}
\begin{longtable}[h!]{l@{\extracolsep{\fill}}p{12cm}}
Name: & propaxiom \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{propaxiom_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{propaxiom_1.00.00_1.00.00.pdf}  \\
Name: & predaxiom \\
Version: & 1.00.00 \\
Rule version: & 1.00.00 \\
Orgin: & \url{predaxiom_1.00.00_1.00.00.qedeq}  \\
pdf: & \url{predaxiom_1.00.00_1.00.00.pdf}  \\
\end{longtable}



First we prove a simple implication:




\begin{thm}
\hypertarget{theo1}{}
\begin{displaymath}
(\forall x(R(x))\ \Rightarrow \  \neg \exists x(\neg R(x)))\end{displaymath}
\end{thm}
\begin{proof}
\mbox{}\\
\begin{longtable}[h!]{r@{\extracolsep{\fill}}p{9cm}@{\extracolsep{\fill}}p{4cm}}
\label{theo1:1}
  $1$ & $(\exists x(R(x))\ \Rightarrow \  R(y))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule2}{add axiom} \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{axiom5}{axiom5} } \\
\label{theo1:2}
  $2$ & $(\exists x(\neg R(x))\ \Rightarrow \  \neg R(y))$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule3}{replace}  $R(@S_{1})$ by $\neg R(@S_{1})$ in \hyperref[theo1:1]{$1$}
} \\
\label{theo1:3}
  $3$ & $(\neg \exists x(\neg R(x))\ \vee \  \neg R(y))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule5}{use abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[theo1:2]{$2$} at occurence $1$
} \\
\label{theo1:4}
  $4$ & $((P\ \vee \  Q)\ \Rightarrow \  (Q\ \vee \  P))$
  & {\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{theo1:5}
  $5$ & $((\neg \exists x(\neg R(x))\ \vee \  Q)\ \Rightarrow \  (Q\ \vee \  \neg \exists x(\neg R(x))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $P$ by $\neg \exists x(\neg R(x))$ in \hyperref[theo1:4]{$4$}
} \\
\label{theo1:6}
  $6$ & $((\neg \exists x(\neg R(x))\ \vee \  \neg R(y))\ \Rightarrow \  (\neg R(y)\ \vee \  \neg \exists x(\neg R(x))))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule4}{replace}  $Q$ by $\neg R(y)$ in \hyperref[theo1:5]{$5$}
} \\
\label{theo1:7}
  $7$ & $(\neg R(y)\ \vee \  \neg \exists x(\neg R(x)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule1}{MP} with \hyperref[theo1:3]{$3$}, \hyperref[theo1:6]{$6$}} \\
\label{theo1:8}
  $8$ & $(R(y)\ \Rightarrow \  \neg \exists x(\neg R(x)))$
  & {\tiny \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{rule6}{reverse abbreviation} \hyperref{propaxiom_1.00.00_1.00.00.pdf}{}{impl}{impl} in \hyperref[theo1:7]{$7$} at occurence $1$
} \\
\label{theo1:9}
  $9$ & $(\forall y(R(y))\ \Rightarrow \  \neg \exists x(\neg R(x)))$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule4}{Particularize} by $y$ in \hyperref[theo1:8]{$8$}} \\
\label{theo1:10}
  $10$ & $(\forall x(R(x))\ \Rightarrow \  \neg \exists x(\neg R(x)))$
  & {\tiny \hyperref{predaxiom_1.00.00_1.00.00.pdf}{}{predrule1}{rename}  $y$ into $x$ in \hyperref[theo1:9]{$9$} at occurence $1$
} \\
\\ & & & \mbox{\qedhere}
\end{longtable}
\end{proof}

\section*{}

\end{document}
