Can we negate a statement this way?

Question

If there is a statement of the form:

$ \forall a$, $ \exists k$ s.t. $ P(x)$ is true

Can we negate it this way:

$ \forall a$, $ \nexists k$ s.t. $P(x)$ is true

Please, tell me if I'm wrong and if there are mutliple ways of negating such a statement.

Thank you.

Answer 1

Correct negation should be, $$\exists a , \text{such that} \; \nexists k \in \Bbb R , P(x) \; \text{is true}$$

Answer 2

$\neg\left(\forall a \, \exists k | P(x)\right)=\exists a \neg(\exists k|P(x))=\exists a \, \forall k|\neg P(x).$