Using compactness theorem for propositional calculus

Question

I am given the following definition:

A set $F$ of sets of propositions is defined "good" if a new satiable set can be constructed by choosing exactly one proposition from each set in $F$.

I need to prove that if all of the sets in $F$ are finite, then

$F$ is good if each finite subset of $F$ is good

My first idea was to in some way merge all of the chosen propositions and then using the compactness theorem but that wouldn't work as each subset of $F$ may require a different choice of propositions. I don't really have a direction to go to prove this so a general direction would help.

thanks.

Answer 1

Hint: If $S=\{\phi_1,\dots,\phi_n\}$ define $$d(S)=\phi_1\lor\dots\lor\phi_n.$$Think about $$\tilde F=\{d(S):S\in F\}.$$