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My question is about the concept of elementary equivalence in logic. According to my book, two structures A and B are elementary equivalent if: for every sentence s: A satisfies s if and only if B satisfies s. However, in my book it is also said that if B satisfies the theory of A, then A and B are elementary equivalent.

It is obvious that if this A satisfies s, then B also satisfies s (since s is in the theory of A). But I don't see how to get the other side of the "if and only if". If B satisfies s, I see no reason for s to be also satisfied by A. If B satisfies the theory of A, B could just as well satisfy other sentences too, right?