Tarski-Vaught test

A substructure $M\subseteq N$ is elementary if and only if, whenever $N\models \exists x\phi (x,\bar{a})$ with parameters $\bar{a}$ from $M$, some witness $b\in M$ satisfies $N\models \phi (b,\bar{a})$.

The witness criterion characterizes an elementary substructure, and hence preservation of all first-order formulas with parameters from the substructure.