Let $\Delta$ be a Skolem theory. Let $M$ be a model of $\Delta$ and $N$ a substructure of $M$. Then we need to proof that for every $L_N$-sentence $\phi$ we have the equivalence $N \models \phi \iff M \models \phi$. By the Tarski-Vaught test it suffices to find an $a \in N$ for all $L_N$-sentences of the form $\exists x \phi(x)$ such that $M \models \phi(a)$. I'm stuck at finding this $a$. Because $\Delta \models \exists x \phi(x) \to\phi(c)$, it would be sufficient to proof that $N \models \Delta$, but I'm not sure whether that is true or how to otherwise approach the problem.

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    $\begingroup$ $N\models\Delta$ is part of the definition of $\Delta$ being model complete. (In other words you don’t need all substructures of $M$ to be elementary, obly those which are also models of $\Delta.$) $\endgroup$ – spaceisdarkgreen Jan 16 at 17:46

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