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\setlength\subproofhorizspace{1.3em}\begin{logicproof}{4} \lnot \exists x \forall y \; (P(x) \land Q(y)) & \prem\\ \begin{subproof} \begin{subproof} \forall y (P(x_0)\land Q(y)) & $\assum$ $\freshVar{$x_0$}$\\ \exists x \forall y (P(x)\land Q(y)) & $\existi2$\\ \bot & $\nege1,3$ \end{subproof} \lnot \forall y (P(x_0)\land Q(y)) & $\negi2-4$\\ \begin{subproof} \lnot \exists y \lnot (P(x_0)\land Q(y)) & $\assum$\\ \begin{subproof} \begin{subproof} \lnot (P(x_0)\land Q(y_0)) & $\assum$ $\freshVar{$y_0$}$\\ \exists y \lnot (P(x_0) \land Q(y)) & $\existi7$\\ \bot & $\nege6,8$ \end{subproof} P(x_0) \land Q(y_0) & $PBC 7-9$ \end{subproof} \forall y (P(x_0)\land Q(y)) & $\foralli7-10$\\ \bot & $\nege5,11$ \end{subproof} \exists y \lnot (P(x_0)\land Q(y)) & $PBC 6-12$ \end{subproof} \forall x \exists y \; \lnot (P(x) \land Q(y)) & $\foralli2-13$\end{logicproof}
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