\begin{logicproof}{2}
  \forall x \; (Q(x) \imp R(x))                  &\prem \\
  \exists x \; (P(x)\land Q(x))                  &\prem \\
  \begin{subproof}
    P(x_0) \land Q(x_0) & \assum~\freshVar{$x_0$}\\
    Q(x_0) \imp R(x_0)  & \foralle 1 $x_0$\\
    P(x_0)              & \ande{1} 3 \\
    Q(x_0)              & \ande{2} 3 \\
    R(x_0)              & \impe 6,4\\
    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 2,3-9
\end{logicproof}