Theorem 2.8 (Tarski-Vaught Test). Assuming that:

Then the following are equivalent:

(i) $h$ is an elementary $L$ -embedding
(ii) For every first order formula $φ (y, x_{1}, \dots, x_{n})$ and every $a_{1}, \dots, a_{n} \in M$ , if there exists $y \in N$ such that $N ⊨ φ (y, h (a_{1}), \dots, h (a_{n}))$ then there exists $y \in M$ such that $N ⊨ φ (h (y), h (a_{1}), \dots, h (a_{n}))$ .