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\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}
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