Complexity of a SAT related problem
Given a propositional formula $\varphi(\vec{x}, \vec{y})$, consider $\psi = \exists \vec{y} \ \forall \vec{x} \ \varphi(\vec{x}, \vec{y})$.

Let 
$\Delta = 
\{ y_0 \leftrightarrow \top, y_0 \leftrightarrow \bot, \ldots, 
y_m \leftrightarrow \top, y_m \leftrightarrow \bot \}$.
Then $\psi$ is true iff there exists $\Gamma \subseteq \Delta$ such that $\Gamma \nvDash \bot$ and $\Gamma \vDash \varphi(\vec{x}, \vec{y})$.

Therefore the problem is $\Sigma^\mathsf{P}_2$-hard.
$\mathsf{DP} \subseteq \Delta^\mathsf{P}_2$, so unless $\mathsf{PH}$ collapses the problem is not in $\mathsf{DP}$.