This sequent is not provable.

Model $\mathcal{M}$:\\
\begin{minipage}{0.2\textwidth}
  \begin{align*}
  \mathcal{A} =& \; \{ a,b \}\\
  P^\mathcal{M} =& \; \{ a,b \}\\
  Q^\mathcal{M} =& \; \{ a \}
  \end{align*}
\end{minipage}\\

$\mathcal{M} \models \forall x \exists y \; (P(x) \imp Q(y)), P(s)$\\
$\mathcal{M} \nmodels \exists x \forall y \; (\lnot P(x) \lor Q(y))$
