I understand that when we want to negate a statement with universal quantifier, that quantifier changes to existential quantifier, and vice versa. For example,
negation of $(\exists x\in\Bbb N)(x+1=8)$ would be $(\forall x\in\Bbb N)(x+1\neq 8)$ and the negation of $(\forall x\in\Bbb N)(x<8)$ would be ($\exists x\in\Bbb N)x\geq 8$.
However, for some reason I don't understand how negating statements with both universal and existential quantifiers works. For instance,
$(\forall x\in\Bbb R)(\exists y\in\Bbb R) x^{2}+y^{2}\geq 4$ should be negated as $(\exists x\in\Bbb R)(\forall y\in\Bbb R) x^{2}+y^{2}<4$. $(*)$
I'm not even sure what I don't understand, I know that
$\neg(\forall x \in\Bbb R)=(\exists x\in\Bbb R)$
$\neg(\exists y\in\Bbb R)=(\forall y\in\Bbb R)$
$\neg(x^{2}+y^{2}\geq 4)=x^{2}+y^{2}<4$
I guess that when reading those statements $(*)$, I don't understand how one negates the other.