How can I prove these quantifier problem? $\neg\forall x P(x) \iff \exists x \neg P(x)$

Question

I am taking a discrete mathematics course in a college now, and studying about the negation of quantifiers. I understood it is correct directly, but want to know how to prove it. Could someone prove it?

There are many proofs of below problems, but I want the proof consists of some statements or sentences like this "$\neg\forall x P(x) \iff \exists x \neg P(x)$".

Below are the problems.

$$ \neg\forall x P(x) \iff \exists x \neg P(x) $$

$$ \neg\exists x P(x) \iff \forall x \neg P(x) $$

Answer 1

Here is a proof of the first equivalence in the Fitch system: