Two equivalent formulas of predicate logic.

Question

I need to find two equivalent formulas of predicate logic F,G s.t in one of them we only have free variables and in the other one we have exactly two quantifiers.

I have the following but this just seems too banal and cheap to me.

$\forall x \neg P(z)$

$\forall x \exists y \neg P(z)$

please help me understand this better as I am stuck. Furthermore, what should I read more to understand this better?

Answer 1

Technically, the formula $P$ already has only free variables (that is, it is true that all of its (zero) free variables are free). So, you could even do:

$P$

And

$\forall x \forall y \ P$