\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}