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