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