# Why is a Skolem theory model complete?

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.

• $N\models\Delta$ 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 $\Delta.$) – spaceisdarkgreen Jan 16 at 17:46