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Consider the following decision problem:

Given: Two (3CNF-)formulas $\varphi_1$, $\varphi_2$ on a shared set $X\cup Y$ of variables ($X$ and $Y$ disjoint).

Question: $\exists$ assignment $\tau_X$ on $X$ such that $\varphi_1$ is satisfiable and $\varphi_2$ is unsatisfiable?

(The "satisfiable" and "unsatisfiable" conditions are relative to the fixed assignment $\tau_X$ from the outer quantifier and can only "choose" the assignment to variables in $Y$.)

This problem is a generalization of the well-known 3-SAT/3-UNSAT problem, which is DP-complete:

Given: Two (3CNF-)formulas $\varphi_1$, $\varphi_2$ on a set $Y$ of variables.

Question: Is $\varphi_1$ satisfiable and $\varphi_2$ unsatisfiable?

The generalization works in the same way in which the NP-complete problem 3-SAT is generalized for higher levels of the polynomial hierarchy. In fact, if the formula $\varphi_1$ and the corresponding condition is removed, this problem coincides with the $\Sigma_2^p$-complete variant of 3-SAT:

Given: A (3CNF-)formula $\varphi_2$ on a set $X\cup Y$ of variables ($X$ and $Y$ disjoint).

Question: $\exists$ assignment $\tau_X$ on $X$ such that $\varphi_2$ is unsatisfiable?

Now I wonder how to call the complexity class that this problem is a member of (and probably complete for). It seems to be $\textit{NP}^{\textit{DP}}$, i.e., $\textit{NP}$ with access to a $\textit{DP}$ oracle, in the same way that $\Sigma_2^p=\textit{NP}^{\textit{coNP}}$ is $\textit{NP}$ with access to a $\textit{coNP}$ oracle. However, I have not found such a class in the literature. Is the problem simply too unnatural to receive any attention? Or can it be simplified such that it falls into another, more common class?

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You have not found explicit mention of it because it is the same class as $\Sigma_2 = \textbf{NP}^\textbf{NP}$. $\Sigma_2 \subseteq \textbf{NP}^\text{DP}$ holds because any $\textbf{NP}$ query can be poly-time (many-one) reduced to a SAT query, which, in turn, can be cast as a DP query (by setting $\varphi_2$ to a trivially unsatisfiable formula). Conversely, $\Sigma_2 \supseteq \textbf{NP}^\text{DP}$ because any DP query can be answered by two (adaptive) SAT queries.

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