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Taking advantage of the model theory, prove that $M \cong N \implies M \equiv N$

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What is the context? Model theory, with $\cong$ as isomorphism and $\equiv$ as elementary equivalence? – Brian M. Scott Jan 22 '13 at 0:27
Yes, that is the context. Forgive me for not mentioning that. – Adam Poterałowicz Jan 22 '13 at 0:28
Please write a full question. Not just a title and some formula. Try to include some context, too. No one here can read minds (I hope). – Asaf Karagila Jan 22 '13 at 0:28
With all due respect, that is the full question. For other notes, please refer to the post just above: "The context is the model theory, with ≅ as isomorphism and ≡ as elementary equivalence." – Adam Poterałowicz Jan 22 '13 at 0:30
This follows almost immediately by the definition of isomorphism and elementary equivalence. What did you try to do so far? Where did you get stuck? – Asaf Karagila Jan 22 '13 at 0:38

HINT: Let $h:M\to N$ be an isomorphism. Show by induction on the complexity of $\varphi(x_1,\dots,x_n)$ that $M\vDash\varphi(a_1,\dots,a_n)$ iff $N\vDash\varphi\big(h(a_1),\dots,h(a_n)\big)$. (By now you should have seen several such proofs.) From this it follows immediately that $M\equiv N$.

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