%% 
%%  Ein Beispiel der DANTE-Edition
%%  Mathematiksatz mit LaTeX
%%  3. Auflage
%%  Beispiel 09-23-1 auf Seite 205.
%%  Copyright (C) 2018 Herbert Voss
%% 
%%  It may be distributed and/or modified under the conditions
%%  of the LaTeX Project Public License, either version 1.3
%%  of this license or (at your option) any later version.
%%  See http://www.latex-project.org/lppl.txt for details.
%% 
%% ==== 
% Show page(s) 1
%% 
%% 
\documentclass[10pt]{exaartplain}
\pagestyle{empty}
\setlength\textwidth{352.81416pt}
\AtBeginDocument{\setlength\parindent{0pt}}
%StartShownPreambleCommands
\usepackage{prooftrees, amsmath, turnstile}
\newcommand*\tnot{\mathord{\sim}}            \newcommand*\lif{\mathbin{\rightarrow}}
\newcommand*\liff{\mathbin{\leftrightarrow}} \newcommand*\elim{\,\text{E}}
%StopShownPreambleCommands
\begin{document}
\begin{prooftree}
  {to prove={(\exists x)((\forall y)(Py \lif x = y) \land Px) \sststile{}{} (\exists
    x)(\forall y)(Py \liff x = y)}}
  [{(\exists x)((\forall y)(Py \lif x = y) \land Px)}, checked=a, just=Pr., name=pr
    [{\lnot (\exists x)(\forall y)(Py \liff x = y)}, subs=a, just=Conc.~neg., name=neg conc
      [{(\forall y)(Py \lif a = y) \land Pa}, checked, just=$\exists\elim$:pr
        [{(\forall y)(Py \lif a = y)}, subs=b, just=$\land\elim$:!u, name=mark
          [Pa, just=$\land\elim$:!uu, name=simple
            [{\lnot (\forall y)(Py \liff a = y)}, checked=b, just=$\lnot\exists\elim$:neg conc
              [{\lnot (Pb \liff a = b)}, checked, just=$\lnot\forall\elim$:!u
                [Pb, just=$\liff\elim$:!u, name=to Pb or not to Pb
                  [a \neq b, just=$\liff\elim$:!u
                  [{Pb \lif a = b}, checked, just=$\forall\elim$:mark, move by=1
                      [\lnot Pb, close={:to Pb or not to Pb,!c}, just=$\lif\elim$:!u
                      ]
                      [{a = b}
                        [a \neq a, close={:!c}, just={$=\elim$:{!uuu,!u}}
                        ]
                      ]
                  ]
                ]
                ]
                [\lnot Pb
                  [{a = b}[Pb, just={$=\elim$:{simple,!u}}, close={:to Pb or not to Pb,!c}]]
  ]]]]]]]]
\end{prooftree}
\end{document}