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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}