\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{1}
    P(x) \lor Q(y)                  &\prem\\
    P(x)\imp R(z)                   &\prem\\
    Q(y) \imp R(z)                  &\prem\\
    \begin{subproof}
        P(x)                        &$\assum$\\
        R(z)                        &$\impe2,4$
    \end{subproof}
    \begin{subproof}
        Q(y)                        &$\assum$\\
        R(z)                        &$\impe3,6$
    \end{subproof}
    R(z)                            &$\ore1,4-5,6-7$
\end{logicproof}