A statement is in prenex normal form (PNF) if and only if it is of the form
$$
Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right)
$$
where each $Q_{i}, i=1,2, \ldots, k,$ is either the existential quantifier or the universal quantifier, and $P\left(x_{1}, \ldots, x_{k}\right)$ is a predicate involving no quantifiers. For example, $\exists x \forall y(P(x, y) \wedge Q(y))$ is in prenex normal form, whereas $\exists x P(x) \vee \forall x Q(x)$ is not (because the quantifiers do not all occur first).
Every statement formed from propositional variables, predicates, $\mathbf{T},$ and $\mathbf{F}$ using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact.
Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section $5.3 . )$
