\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{1}
  \exists x \;(P(x) \land Q(x))                     & prem.\\
  \begin{subproof}
    \llap{$x_0\enspace \;$} P(x_0) \land Q(x_0)     & ass.\\
    P(x_0)                                          & $\land\mathrm{e} 2$\\
    Q(x_0)                                          & $\land\mathrm{e} 2$\\
    \exists x \; P(x)                               & $\exists\mathrm{i} 3$\\
    \exists x \; Q(x)                               & $\exists\mathrm{i} 4$\\
    \exists x \; P(x) \land \exists \; Q(x)         & $\land\mathrm{i} 5,6$
  \end{subproof}
  \exists x \;P(x) \land \exists x \; Q(x))         & $\exists\mathrm{e} 1,2-7$
\end{logicproof}