I was wondering what the general rules for negating quantifiers was.
I was reading that they follow this rule holds:
$$NOT(\forall x. P(x)) \iff \exists x. NOT(P(x))$$
Which makes sense to me. However, I was worried that if there are more and more predicates and quantifiers involved in the negation, that we have to be extra careful about negating the statement and that its not as simple as just "distributing" the NOT.
For example is just "distributing" the NOT over all predicates and quantifiers really negates a statement correct?
Consider this example:
$$NOT( \exists x. P(x). \forall y. P(x,y). \exists z \exists k. P(z,k)) \iff \forall x. NOT(P(x)). \exists y. NOT(P(x,y)). \forall z \forall k. NOT(P(z,k))$$
Basically, is it safe to just distribute the NOT across all quantifiers and predicates when trying to negate a more general proposition? The order of the statements in the proposition never change when we negate, right? Is there some intuitive proof that distribution of not works for negating a proposition? Why is "distributing" the not correct? Or maybe someone could make the rule precise to me, its not clear to me how to do it for an arbitrary proposition with crazy mixtures of quantifiers and predicates.