\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{4}
    \lnot \exists x \forall y \; (P(x) \land Q(y))              & \prem\\
    \begin{subproof}
        \begin{subproof}
            \forall y (P(x_0)\land Q(y))                        & $\assum$ $\freshVar{$x_0$}$\\
            \exists x \forall y (P(x)\land Q(y))                 & $\existi2$\\
            \bot                                                & $\nege1,3$
        \end{subproof}
        \lnot \forall y (P(x_0)\land Q(y))                      & $\negi2-4$\\
        \begin{subproof}
            \lnot \exists y \lnot (P(x_0)\land Q(y))            & $\assum$\\
            \begin{subproof}
                \begin{subproof}
                    \lnot (P(x_0)\land Q(y_0))                  & $\assum$ $\freshVar{$y_0$}$\\
                    \exists y \lnot (P(x_0) \land Q(y))          & $\existi7$\\
                    \bot                                        & $\nege6,8$
                \end{subproof}
                P(x_0) \land Q(y_0)                             & $PBC 7-9$
            \end{subproof}
            \forall y (P(x_0)\land Q(y))                        & $\foralli7-10$\\
            \bot                                                & $\nege5,11$
        \end{subproof}
        \exists y \lnot (P(x_0)\land Q(y))                      & $PBC 6-12$
    \end{subproof}
    \forall x \exists y \; \lnot (P(x) \land Q(y))              & $\foralli2-13$
\end{logicproof}