\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{2}
    P(y) \imp \forall x Q(x)                & \prem\\
    \exists x \lnot Q(x)                    & \prem\\
    \begin{subproof}
        P(y)                                & $\assum$\\
        \forall x Q(x)                      & $\impe1,3$\\
        \begin{subproof}
            \lnot Q(x_0)                    &$\assum$ $\freshVar{$x_0$}$\\
            Q(x_0)                          &$\foralle4$\\
            \bot                            &$\nege5,6$
        \end{subproof}
        \bot                                &$\existe2,5-7$
    \end{subproof}
    \lnot P(y)                              &$\negi3-8$\\
    \exists x \lnot P(x)                    &$\existi9$
\end{logicproof}