Questionnaire-2024/natural_deduction_predicate_logic/9999_sol.tex

\begin{logicproof}{1}
  \forall x \; (P(x) \land Q(x))                   &prem \\
  \begin{subproof}
    \llap{$x_0\enspace \;$} \: P(x_0) \land Q(x_0) &$\forall \mathrm{e} 1 $\\
    P(x_0)                                         &$\land \mathrm{e} 2$
  \end{subproof}
  \forall x \; P(x)                                &$\forall \mathrm{i} 2-3 $\\
  \begin{subproof}
    \llap{$x_1\enspace \;$} \: P(x_1) \land Q(x_1) &$\forall \mathrm{e} 1 $\\
    Q(x_1)                                         &$\land \mathrm{e} 5$
  \end{subproof}
  \forall x \; Q(x)                                &$\forall \mathrm{i} 5-6 $\\
  \forall x \; P(x) \land \forall x \; Q(x)        &$\land \mathrm{i} 4,7$
  \end{logicproof}