\setlength\subproofhorizspace{1.1em}
\begin{logicproof}{1}
  \forall a \forall b \; (P(a) \land Q(b))                  & prem.\\
  \begin{subproof}
	  \llap{$t\enspace \;$} \forall b \; (P(s) \land Q(b))    & $\forall \mathrm{e}$ 1\\
    P(s) \land Q(t)                                         & $\forall \mathrm{e}$ 2\\
    P(s)                                                    & $\land \mathrm{e}_1$ 3\\
    P(s) \lor Q(t)                                          & $\lor \mathrm{i}_1$ 4\\
    \exists b \; (P(s) \lor Q(b))                           & $\exists \mathrm{i}$ 5
  \end{subproof}
  \forall a \exists b \; (P(a) \lor Q(b))                   & $\forall \mathrm{i}$ 2-6
\end{logicproof}