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))))

Firstly, negate your quantifiers.
Negation of Universal quantifier:

$$\neg \forall s: S(s) \quad\equiv\quad \exists s: \neg S(s)$$
Negation of Existential quantifier:

$$\neg \exists s: S(s) \quad\equiv\quad \forall s: \neg S(s)$$ 
Note that these iterate, thusly:
 $$\neg \forall x \exists y \forall z : S(x,y,z) \quad\equiv\quad \exists x\forall y \exists z: \neg S(x,y,z)$$

Next negate your predicated statements, by applying Implication Equivalence and  DeMorgan's Laws repeatedly to each bracketed expression until all negations are before a predicate.
Material Implication Equivalence:

$$(S\to T) \quad\equiv\quad (\neg S\lor T)$$
Negation of Conjunction: 

$$\neg (S\land T) \quad\equiv\quad (\neg S \lor \neg T)$$
Negation of Disjunction: 

$$\neg (S\lor T) \quad\equiv\quad (\neg S \land \neg T)$$
