\setlength\subproofhorizspace{1em}
\begin{logicproof}{0}
  \forall x \; P(x) \land \forall x \; (P(y) \imp Q(x))     & prem.\\
  \forall x \; P(x)                                         & $\land \mathrm{e}_1$ 1\\
  \forall x \; (P(y) \imp Q(x))                             & $\land \mathrm{e}_2$ 1\\
  P(y)                                                      & $\forall \mathrm{e}$ 2\\
  P(y) \imp Q(z)                                            & $\forall \mathrm{e}$ 3\\
  Q(z)                                                      & $\imp \mathrm{e}$ 5,4
\end{logicproof}