
\begin{logicproof}{2}
  \forall x \; (P(x) \land Q(x))                   &\prem \\
  \begin{subproof}
                             & \freshVar{$x_0$} \\
    P(x_0) \land Q(x_0)  & \foralle 1 $x_0$ \\  
    P(x_0)                 & \ande{1}1 \\
    Q(x_0)                 & \ande{2}1 \\
    R(x_0) \lor P(x_0)         & \ori 4 \\
    Q(x_0) \lor R(x_0) & \ori 5 \\
    (Q(x_0) \lor R(x_0)) \land (R(x) \lor P(x)) & \andi 6,7
  \end{subproof}
  \forall x \; ((Q(x) \lor R(x)) \land (R(x) \lor P(x))) & \foralli 2-8
\end{logicproof}
