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\setlength\subproofhorizspace{1.3em}\begin{logicproof}{1} \exists x \;(P(x) \land Q(x)) & prem.\\ \begin{subproof} \llap{$x_0\enspace \;$} P(x_0) \land Q(x_0) & ass.\\ P(x_0) & $\land\mathrm{e} 2$\\ Q(x_0) & $\land\mathrm{e} 2$\\ \exists x \; P(x) & $\exists\mathrm{i} 3$\\ \exists x \; Q(x) & $\exists\mathrm{i} 4$\\ \exists x \; P(x) \land \exists \; Q(x) & $\land\mathrm{i} 5,6$ \end{subproof} \exists x \;P(x) \land \exists x \; Q(x)) & $\exists\mathrm{e} 1,2-7$\end{logicproof}
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