This is part 2 of a question I asked here:
Prove this claim about language and structures.
The setting is that suppose $\phi_1,\ldots,\phi_n$ are $\mathcal{L}$-formulas and $\psi$ is a Boolean combination of $\phi_1,\ldots,\phi_n$. Then we showed there is $S \subseteq \mathcal{P}(\{1,\ldots,n\})$ such that
$$\models \psi \Leftrightarrow \bigvee_{X \in S} \left(\bigwedge_{i \in X} \phi_i \wedge \bigwedge_{i \notin X} \neg \phi_i \right)$$
Now I would like to show that every formula is equivalent to one of the form
$$Q_1 v_1 \ldots Q_m v_m ~~\psi$$
Where $\psi$ is quantifier free and each $Q_i$ is either $\forall$ or $\exists$.
First, intuitively speaking, what is this question asking? In addition to proving this claim, I would like to know if there is any online reading material associated with what this question is asking, so I may read into it some more.