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I'll leave the question here in case someone else had an issue. The variables of the formulas are evaluated in $M$. Clarifies everything!

From my understanding of the Vaught Tarsi test, it states the following: Suppose $M\subseteq N$ is a substructure of $N$. Then $M \preceq N$ (i.e. a formula is true in $M$ if and only if it is true in $N$) if and only if for all formulas $\varphi$ of the form $\varphi = \exists x \psi$, $$ N \vDash \varphi \implies \text{ there exists } a \in M \textit{ such that } N \vDash \psi(a) $$ The general proof is inductive. But my confusion is about the universal quantifier. If a formula is of the form $\forall\psi$ then couldn't it be true in $M$ without being true in $N$?

I am a beginner and possibly misunderstanding this a lot, so any explanation is welcome.

Thank you!

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    I don't understand the vote to close - this seems perfectly clear to me . . . – 2017-02-17
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    The vote to close reflects one moment in time in the history of this question. It's still not really clear what a question that now includes "Clarifies everything!" in the first paragraph is actually about. – 2017-02-17

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Any formula is equivalent to a formula with only existential quantifiers, since you can replace every quantifier $\forall x$ with $\neg\exists x \neg$. So in proving $M\preceq N$, you can consider only formulas without universal quantifiers.

Another way to think about what's going on is that if $\forall x:\varphi(x)$ were not true in $N$, then that means $\exists x:\neg\varphi(x)$ is true in $N$. By the assumption, this existential statement must have a witness in $M$. As a result, $\forall x:\varphi(x)$ will end up being false in $M$ as well.