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