This sequent is provable.

\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{2}
    \exists y \forall x P(x,y) & \prem \\
    \begin{subproof}
        \forall x P(x,y_0) & $\assum$ $\freshVar{$y_0$}$ \\
        \begin{subproof}
            P(x_0,y_0) & $\foralle 2$ $\freshVar{$x_0$}$ \\
            \exists y P(x_0,y) & $\existi 3$
        \end{subproof}
        \forall x \exists y P(x,y) & $\foralli3-4$
    \end{subproof}
    \forall x \exists y P(x,y) & $\existe 1,2-5$
\end{logicproof}