A compactness problem for model theory I'm working on the following problem:
Assume that every model of a sentence $\varphi$ satisfies a sentence from $\Sigma$. Show that there is a finite $\Delta \subseteq \Sigma$ such that every model of $\varphi$ satisfies a sentence in $\Delta$.
The quantifiers in this problem are throwing me off; besides some kind of compactness application I'm not sure where to go with it (hence the very poor title). Any hint?

Cute, in a twisted sort of way.  You are right, the  quantifier structure is the main hurdle to solving the problem. 
We can assume that $\varphi$ has a model, else the result is trivially true.
Suppose that there is no finite $\Delta\subseteq \Sigma$ with the desired property.
Then for every finite  $\Delta \subseteq \Sigma$, the set $\{\varphi, \Delta'\}$ has a model.  (For any set $\Gamma$ of sentences, $\Gamma'$ will denote  the set of negations of sentences in $\Gamma$.)
By the Compactness Theorem, we conclude that $\{\varphi, \Sigma'\}$ has a model $M$.
This model $M$ is a model of $\varphi$ in which no sentence in $\Sigma$ is true, contradicting the fact that every model of $\varphi$ satisfies a sentence from $\Sigma$.
