If I had the following logical expression, how would I simplify it? I need to be sure that negations appear only applied to predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). In the expression below, ~ means "not," $\displaystyle \vee$ means "or," and $\displaystyle \wedge$ means "and."

$\displaystyle \sim \left[ \forall x ( \exists y \forall z P(x,y,z) \wedge \exists z \forall y P(x,y,z) \right] $

I'm fairly certain that the not can be distributed inwards to achieve:

$\displaystyle \left[ \sim \left( \forall x ( \exists y \forall z P(x,y,z)) \right] \vee \left[ \sim (\exists z \forall y P(x,y,z) \right) \right] $

... but I am not sure how to proceed from there.
Thanks in advance!