= Solution
The <Tarski-Vaught test> says that a <substructure of a first-order structure>[substructure] $A\subseteq B$ is elementary exactly when for every formula $\phi(x,\bar y)$ and tuple $\bar a\in A$, if $B\models\exists x\phi(x,\bar a)$, some $b\in A$ satisfies $B\models\phi(b,\bar a)$. Thus \b[every existential assertion with parameters from $A$ that has a witness in $B$ already has a witness in $A$]. The matrix may contain <quantifiers>; restricting it to <quantifier-free formulas> would not state the full test.
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