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\setlength\subproofhorizspace{1.6em}
\begin{logicproof}{1}
\exists x \; \lnot P(x) & prem.\\
\forall x \; \lnot Q(x) & prem.\\
\begin{subproof}
\llap{$x_0\enspace \;$} \lnot P(x_0) & ass.\\
\lnot Q(x_0) & $\forall \mathrm{e}$ 2\\
\lnot P(x_0) \land \lnot Q(x_0) & $\land \mathrm{i}$ 3,4\\
\exists x \; (\lnot P(x) \land \lnot Q(x)) & $\exists \mathrm{i}$ 5
\end{subproof}
\exists x \; (\lnot P(x) \land \lnot Q(x)) & $\exists \mathrm{e}$ 3-6
\end{logicproof}