Universal theory and substructures of elementarily equivalent models ($T\cup {\mathrm{Th}}_{\forall }(A)\text{ satisfiable}⟺B\models T,B\subseteq A^{\ast }\equiv A$)

If $B$ satisfies the universal theory of $A$, every existential sentence true in $B$ is true in $A$, since its falsehood in $A$ would supply a universal negation. Apply the existential-theory embedding criterion to $B$ and the complete theory of $A$ to obtain $A^{\ast }$. Conversely universal truth descends to substructures. The elementarily equivalent model need not be the original structure itself.