\setlength\subproofhorizspace{1em}
\begin{logicproof}{0}
  \forall x \; (P(x) \imp Q(y))     & prem.\\
  \forall y \; (P(y) \land R(x))    & prem.\\
  P(t) \imp Q(y)                    & $\forall \mathrm{e}$ 1\\
  P(t) \land R(x)                  & $\forall \mathrm{e}$ 2\\
  P(t)                              & $\land \mathrm{e}_1$ 4\\
  Q(y)                              & $\imp \mathrm{e}$ 3\\
  \exists x \; Q(x)                 & $\exists \mathrm{i}$ 6
\end{logicproof}