Questionnaire-2024/natural_deduction_predicate_logic/0008.tex

\item  \ifassignmentsheet \points{4} \fi Consider the following natural deduction proof for the sequent $$\forall x \; (P(x)\imp Q(x)), \quad \exists x \; P(x) \quad \ent \quad \forall x Q(x).$$
Is the proof correct? If not, explain the error in the proof and either show how to correctly prove the sequent, or give a counterexample that proves the sequent invalid.

\setlength\subproofhorizspace{1.1em}
\begin{logicproof}{2}
  \forall x \; (P(x)\imp Q(x))      & prem.\\
  \exists x \; P(x)                 & prem.\\
  \begin{subproof}
    \llap{$x_0\enspace \;$}		      &\\
    \begin{subproof}
      P(x_0)                        & ass.\\
      P(x_0) \imp Q(x_0)            & $\forall \mathrm{e}$ 1\\
      Q(x_0)                        & $\imp \mathrm{e}$, 4,5
    \end{subproof}
    \forall x \; Q(x)               & $\forall \mathrm{i}$ 4-6
  \end{subproof}
  \forall x \; Q(x)                 & $\exists \mathrm{e}$ 2,3-7
\end{logicproof}