This sequent is not provable.

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

$\mathcal{M} \models \exists x \; P(x) \imp Q(x)$\\
$\mathcal{M} \models \lnot Q(z)$\\
$\mathcal{M} \nmodels \lnot P(z)$