This sequent is not provable.

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

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