Changing the ∀ quantifier to the ∃ quantifier

I am tasked with changing the $\forall$ quantifierwith the $\exists$ quantifier. However, I am not sure, whether I understood it correctly, and would like to ask for help.

As I understood, the overarching rule is: $\forall x\in\mathbb{Z}:p(x)$is equal to$\neg(\exists x \in\mathbb{Z}):\neg(p(x))$

Applying this rule to two examples:

1. $\exists x\in\mathbb{N}$,$\forall y\in\mathbb{Z}$: x-y²x=0. When replacing the $\forall$ quantifier: $\exists x\in\mathbb{N}$,$\neg(\exists y\in\mathbb{Z})$: x-y²x≠0. Further, this statement is True for x = 0

2. $\forall x\in\mathbb{N}$,$\forall y\in\mathbb{Z}$ : $yx\in\mathbb{N}$. When replacing the $\forall$ quantifier two times: $\neg(\exists x\in\mathbb{N}$),$\neg(\exists y\in\mathbb{Z}$): $\neg(yx\in\mathbb{N}$). Further, this statement is false for all y < 0.

Especially, in the second example I am unsure whether two replacements will result in $\neg(yx\in\mathbb{N}$) or$(yx\in\mathbb{N}$)

Thanks!