Tarski-Vaught Test

A syntactic criterion to determine when a substructure A of a structure M (in a given first-order language) is an elementary substructure: A is elementary in M iff for every formula φ(x,y) and every tuple a from A, if M satisfies ∃x φ(x,a) then there exists b in A such that M satisfies φ(b,a). Equivalently, A is closed under existential witness-finding in M for formulas with parameters from A.