\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{2}
    \forall x \exists y \; \lnot (P(x) \land Q(y))          & \prem\\
    \exists y \lnot (P(x_0)\land Q(y))                       & $\foralle1$\\
    \begin{subproof}
        \lnot (P(x_0)\land Q(y_0))                          & $\assum$ $\freshVar{$y_0$}$\\
        \begin{subproof}
            \forall y (P(x_0)\land Q(y))                    & $\assum$\\
            P(x_0)\land Q(y_0)                              & $\foralle4$\\
            \bot                                            & $\nege3,5$
        \end{subproof}
        \lnot \forall y (P(x_0)\land Q(y))                  & $\negi4-6$
    \end{subproof}
    \lnot \forall y (P(x_0)\land Q(y))                      & $\existe2,3-7$\\
    \begin{subproof}
        \exists x \forall y (P(x)\land Q(y))                 & $\assum$\\
        \begin{subproof}
            \forall y (P(x_0)\land Q(y))                    & $\assum$ $\freshVar{$x_0$}$\\
            \bot                                            & $\nege8,10$
        \end{subproof}
        \bot                                                & $\existe9,10-11$
    \end{subproof}
    \lnot \exists x \forall y \; (P(x) \land Q(y))          & $\negi9-12$
\end{logicproof}