This sequent is not provable.

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

$\mathcal{M} \models \lnot \exists x \lnot P(x)$\\
$\mathcal{M} \nmodels \forall x \lnot P(x)$