How to simplify this logical expression Full Disclaimer, this is a homework problem. 
Negate the following logical expression and transform it so that negations
only appear before individual predicates:
∀x∃y∀z, P(x) ∧ ¬Q(x) → R(x) ∨ (R(y) ∧ ¬Q(z))
I'm not sure how to start with this problem. I thought that it would make sense to try to bring out the individual not's to the front but I'm not sure because I only know a handful of basic negations like p → q = ¬p ∨ q, which I don't know if it would work here.
I put parentheses afterwords to more clearly highlight the order of precedence:
∀x∃y∀z, ((P(x) ∧ ¬Q(x)) → (R(x) ∨ (R(y) ∧ ¬Q(z))))

What's the negation of '$\forall x \phi$? Let's recall the following first-order logical equivalences. 
For the universal quantifier:

$\neg \forall x \phi \equiv \exists x \neg \phi$

For the existencial one:

$\neg \exists x \phi \equiv \forall x \neg \phi$

So, can you continue from there? In case of any doubt leave a comment.
