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\documentclass[landscape]{article}
\usepackage[a4paper,margin=0.5in,landscape]{geometry}
\newcommand{\Tarrowintro}{$\rightarrow$I}
\newcommand{\Tarrowelim}{$\rightarrow$E}
\newcommand{\Tconjintro}{$\land$I}
\newcommand{\Tconjelim}{$\land$E}
\newcommand{\Tdisjintro}{$\lor$I}
\newcommand{\Tdisjelim}{$\lor$E}
\newcommand{\Tunivintro}{$\forall$I}
\newcommand{\Tunivelim}{$\forall$E}
\newcommand{\Texistintro}{$\exists$I}
\newcommand{\Texistelim}{$\exists$E}
\newcommand{\Tarrow}{\rightarrow}
\newcommand{\Tand}{\land}
\newcommand{\Tor}{\lor}
\newcommand{\Tforall}{\forall}
\newcommand{\Texists}{\exists}
\newcommand{\Tneg}{\lnot}
\usepackage{bussproofs}
\begin{document}
\begin{prooftree}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{\left(A \Tor \Tneg{A}\right)}} \Tarrow \left(A \Tor \Tneg{A}\right)$}
\AxiomC{[$\Tneg{\left(A \Tor \Tneg{A}\right)}$]}
\AxiomC{[$\Tneg{\left(A \Tor \Tneg{A}\right)}$]}
\AxiomC{[$A$]}
\RightLabel{\Tdisjintro}
\UnaryInfC{$A \Tor \Tneg{A}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{A}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$A \Tor \Tneg{A}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{\left(A \Tor \Tneg{A}\right)}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$A \Tor \Tneg{A}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{DP}
\UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\AxiomC{[$\Tneg{\Tforall_{x} P{x}}$]}
\AxiomC{[$P{x} \Tarrow \Tforall_{x} P{x}$]}
\AxiomC{[$P{x}$]}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Tforall_{x} P{x}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{P{x}}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \Tneg{P{x}}$}
\RightLabel{\Texistelim}
\BinaryInfC{$\Texists_{x} \Tneg{P{x}}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tforall_{x} P{x}} \Tarrow \Texists_{x} \Tneg{P{x}}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{DP}
\UnaryInfC{$\Texists_{x} \left(\left(P{x} \Tarrow Q{x}\right) \Tarrow \Tforall_{x} \left(P{x} \Tarrow Q{x}\right)\right)$}
\AxiomC{[$\Tneg{\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)}$]}
\AxiomC{[$\left(P{x} \Tarrow Q{x}\right) \Tarrow \Tforall_{x} \left(P{x} \Tarrow Q{x}\right)$]}
\AxiomC{[$P{x} \Tarrow Q{x}$]}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\left(P{x} \Tarrow Q{x}\right)}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$}
\RightLabel{\Texistelim}
\BinaryInfC{$\Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tforall_{x} \left(P{x} \Tarrow Q{x}\right)} \Tarrow \Texists_{x} \Tneg{\left(P{x} \Tarrow Q{x}\right)}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{DP}
\UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\AxiomC{[$\Tforall_{x} \left(P{x} \Tor Q{x}\right)$]}
\RightLabel{\Tunivelim}
\UnaryInfC{$P{x} \Tor Q{x}$}
\AxiomC{[$P{x} \Tarrow \Tforall_{x} P{x}$]}
\AxiomC{[$P{x}$]}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Tforall_{x} P{x}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$}
\AxiomC{[$Q{x}$]}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} Q{x}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$}
\RightLabel{\Tdisjelim}
\TrinaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$}
\RightLabel{\Texistelim}
\BinaryInfC{$\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tforall_{x} \left(P{x} \Tor Q{x}\right) \Tarrow \left(\Tforall_{x} P{x} \Tor \Texists_{x} Q{x}\right)$}
\end{prooftree}
\clearpage
\begin{prooftree}
\AxiomC{}
\RightLabel{GLPO}
\UnaryInfC{$\Tforall_{x} \Tneg{A} \Tor \Texists_{x} A$}
\AxiomC{[$\Tforall_{x} \Tneg{A}$]}
\RightLabel{\Tunivelim}
\UnaryInfC{$\Tneg{A}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$A \Tor \Tneg{A}$}
\AxiomC{[$\Texists_{x} A$]}
\AxiomC{[$A$]}
\RightLabel{\Texistelim}
\BinaryInfC{$A$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$A \Tor \Tneg{A}$}
\RightLabel{\Tdisjelim}
\TrinaryInfC{$A \Tor \Tneg{A}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{GLPO}
\UnaryInfC{$\Tforall_{x} \Tneg{P{y}} \Tor \Texists_{x} P{y}$}
\AxiomC{[$\Tforall_{x} \Tneg{P{y}}$]}
\RightLabel{\Tunivelim}
\UnaryInfC{$\Tneg{P{y}}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\AxiomC{[$\Texists_{x} P{y}$]}
\AxiomC{[$P{y}$]}
\RightLabel{\Texistelim}
\BinaryInfC{$P{y}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\RightLabel{\Tdisjelim}
\TrinaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\RightLabel{\Tunivintro}
\UnaryInfC{$\Tforall_{y} \left(P{y} \Tor \Tneg{P{y}}\right)$}
\RightLabel{\Tunivelim}
\UnaryInfC{$P{x} \Tor \Tneg{P{x}}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{GLPO}
\UnaryInfC{$\Tforall_{x} \Tneg{P{y}} \Tor \Texists_{x} P{y}$}
\AxiomC{[$\Tforall_{x} \Tneg{P{y}}$]}
\RightLabel{\Tunivelim}
\UnaryInfC{$\Tneg{P{y}}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\AxiomC{[$\Texists_{x} P{y}$]}
\AxiomC{[$P{y}$]}
\RightLabel{\Texistelim}
\BinaryInfC{$P{y}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\RightLabel{\Tdisjelim}
\TrinaryInfC{$P{y} \Tor \Tneg{P{y}}$}
\end{prooftree}
\vspace{1cm}
\begin{prooftree}
\AxiomC{}
\RightLabel{GLPO}
\UnaryInfC{$\Tforall_{x} \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)} \Tor \Texists_{x} \left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)$}
\AxiomC{[$\Tforall_{x} \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$]}
\RightLabel{\Tunivelim}
\UnaryInfC{$\Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$}
\AxiomC{[$\Texists_{x} \left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)$]}
\AxiomC{[$P{y} \Tand \Tneg{\Tforall_{x} P{x}}$]}
\RightLabel{\Texistelim}
\BinaryInfC{$P{y} \Tand \Tneg{\Tforall_{x} P{x}}$}
\RightLabel{\Tdisjintro}
\UnaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$}
\RightLabel{\Tdisjelim}
\TrinaryInfC{$\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$}
\RightLabel{\Tunivintro}
\UnaryInfC{$\Tforall_{y} \left(\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{y} \Tand \Tneg{\Tforall_{x} P{x}}\right)}\right)$}
\RightLabel{\Tunivelim}
\UnaryInfC{$\left(P{x} \Tand \Tneg{\Tforall_{x} P{x}}\right) \Tor \Tneg{\left(P{x} \Tand \Tneg{\Tforall_{x} P{x}}\right)}$}
\end{prooftree}
\clearpage
\begin{prooftree}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}} \Tarrow \Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\AxiomC{[$\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}$]}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{P{x}}} \Tarrow P{x}$}
\AxiomC{[$\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}$]}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{\Tforall_{x} P{x}}} \Tarrow \Tforall_{x} P{x}$}
\AxiomC{[$\Tneg{P{x}}$]}
\AxiomC{[$P{x}$]}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{\Tforall_{x} P{x}}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Tforall_{x} P{x}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$P{x} \Tarrow \Tforall_{x} P{x}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{P{x}}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$P{x}$}
\RightLabel{\Tunivintro}
\UnaryInfC{$\Tforall_{x} P{x}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$P{x} \Tarrow \Tforall_{x} P{x}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Texists_{x} \left(P{x} \Tarrow \Tforall_{x} P{x}\right)$}
\end{prooftree}
\begin{prooftree}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}} \Tarrow \Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$}
\AxiomC{[$\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}$]}
\AxiomC{}
\RightLabel{DNE}
\UnaryInfC{$\Tneg{\Tneg{P{x}}} \Tarrow P{x}$}
\AxiomC{[$\Texists_{x} P{x}$]}
\AxiomC{[$\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}$]}
\AxiomC{[$P{x}$]}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Texists_{x} P{x} \Tarrow P{x}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Texistelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{P{x}}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$P{x}$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Texists_{x} P{x} \Tarrow P{x}$}
\RightLabel{\Texistintro}
\UnaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\bot$}
\RightLabel{\Tarrowintro}
\UnaryInfC{$\Tneg{\Tneg{\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)}}$}
\RightLabel{\Tarrowelim}
\BinaryInfC{$\Texists_{x} \left(\Texists_{x} P{x} \Tarrow P{x}\right)$}
\end{prooftree}
\end{document}
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