\setlength\subproofhorizspace{1.7em}
\begin{logicproof}{2}
  \forall x \; P(x) \lor \forall x \; Q(x)     & prem.\\
  \begin{subproof}
    \forall x \; P(x)                          & ass.\\
    \begin{subproof}
      \llap{$t\enspace \;$} P(t)               & $\forall \mathrm{e}$ 2\\
      P(t) \lor Q(t)                           & $\lor \mathrm{i}_1$ 3
    \end{subproof}
    \forall y \; (P(y) \lor Q(y))               & $\forall \mathrm{i}$ 3-4
  \end{subproof}
  \begin{subproof}
    \forall x \; Q(x)                          & ass.\\
    \begin{subproof}
      \llap{$s\enspace \;$} Q(s)               & $\forall \mathrm{e}$ 6\\
      P(s) \lor Q(s)                           & $\lor \mathrm{i}_2$ 7
    \end{subproof}
    \forall y \; (P(y) \lor Q(y))               & $\forall \mathrm{i}$ 7-8
  \end{subproof}
 \forall y \; (P(y) \lor Q(y))                       & $\lor \mathrm{e}$ 1,2-5,6-9
\end{logicproof}