Łoś-Tarski preservation theorem ($T\equiv T_{\forall }$)

A first-order theory has its class of first-order models closed under substructures of a first-order structure exactly when it is equivalent to its universal consequences $T_{\forall }$. If $A\models T_{\forall }$, then $T\cup \mathrm{Diag}(A)$ is consistent. Otherwise a finite logical conjunction $\delta$ of signed diagram of a structure sentences yields a universal consequence $\forall \mathbf{x}\neg \delta (\mathbf{x})$ false in $A$. The compactness theorem gives a first-order model of $T$ containing an embedded copy of $A$, and closure under substructures of a first-order structure implies $A\models T$. The converse is preservation of universal sentences. The diagram of a structure must include negated as well as positive atomic sentences so that the resulting map is an embedding, not just a homomorphism.

A first-order model class axiomatized by a theory is closed under extensions exactly when it has an existential axiomatization. If $A$ satisfies all existential consequences of $T$, then $T$ is jointly satisfiable with the universal theory of $A$: any finite obstruction would negate a universal conjunction and produce a violated existential consequence. Embed a model of $T$ into a model elementarily equivalent to $A$, apply extension closure, and transfer truth back to $A$.