
\begin{logicproof}{2}
  \forall x \; (P(x) \lor Q(x))                   &\prem \\
  \forall x \; (\lnot P(x))                       &\prem \\
  \begin{subproof}
    & \freshVar{$x_0$}\\
    P(x_0) \lor Q(x_0)                            &$\foralle 1$  $x_0$  \\
    \lnot P(x_0)                                  &$\foralle 2$ \\
    \begin{subproof}
      P(x_0)                                        &\assum \\
      \bot                                          &$\nege 5,6$ \\
      Q(x_0)                                        &$\bote 7$
    \end{subproof}
    \begin{subproof}
      Q(x_0)                                        &\assum
    \end{subproof}
    Q(x_0)                                            &$\ore 4,6-8,9$
  \end{subproof}
  \forall x \; Q(x)                                &$\foralli 3-10 $
\end{logicproof}
