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\begin{logicproof}{0}
\forall x \; (P(x) \land Q(x)) &prem \\
P(x_0) \land Q(x_0) &$\forall \mathrm{e} 1$ \\
P(x_0) &$\land \mathrm{e} 2$ \\
P(x_0) \lor Q(x_0) &$\lor \mathrm{i} 3$ \\
\exists x \; (P(x) \lor Q(x)) &$\exists \mathrm{i} 4$
\end{logicproof}