Existential-theory embedding criterion ($T\cup {\mathrm{Th}}_{\exists }(A)\text{ satisfiable}⟺A\hookrightarrow B\models T$)

A finite conjunction from the signed atomic diagram of $A$ becomes, after existentially closing its new names, a sentence in its existential theory. Joint satisfiability with $T$ therefore makes the full diagram finitely satisfiable with $T$. Compactness gives the embedding, with inequalities ensuring injectivity and signed relation literals ensuring reflection. The reverse implication is preservation of existential truth under embeddings.

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.