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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}
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