\begin{logicproof}{2}
  \exists x \; (Q(x) \imp R(x))                   &\prem \\
  \exists x \; (P(x)\land Q(x))                   &\prem \\
  \begin{subproof}
    Q(x_0) \imp R(x_0)              &\assum~\freshVar{$x_0$} \\
    \begin{subproof}
      P(x_0) \land Q(x_0)             &\assum~\freshVar{$x_0$}\\
      P(x_0)                          &\ande{1} 4\\
      Q(x_0)                          &\ande{2} 4\\
      R(x_0)                          &\impe 5,3\\
      P(x_0) \land R(x_0) & \andi 5,7\\
      \exists x (P(x) \land R(x)) & \existi 8
    \end{subproof}
    \exists x \; (P(x) \land R(x)) & \existe 2,4-9
  \end{subproof}
  \exists x \; (P(x) \land R(x)) & \existe 1,3-10
\end{logicproof}