In general, are the statements "$\exists x,y \in \mathbb{R}$" and "$\forall x,y \in \mathbb{R}$" the exact same as "$\exists x \in \mathbb{R} \exists y \in \mathbb{R}$?" and "$\forall x \in \mathbb{R} \forall y \in \mathbb{R}$," respectively?
If so, why is this the case? If not, how do they differ? For example, if I were to find the negation of "$\forall x,y P(x,y)$," would it be the same as applying the single-variable case $\neg(\forall x P(x)) = \exists x \neg P(x)$? Or is there a different rule? How would it extend to quantifiers of arbitrarily many variables, like $\forall w,x,y,z,\cdots P(w,x,y,z,\cdots)$? Or, maybe, does one statement allow you to let $y$ depend on $x$ while another does not? I'm rather confused on all of these notions and similar ones.
Thank you in advance!
$\exists x \in A: \phi(x)$ is a standard abbreviation for $\exists x: (x \in A) \land \phi(x)$.
$\forall x,y \in A: \phi(x,y)$ similarly means $\forall x: \forall y (x \in A \land y \in A) \to \phi(x,y)$ etc.
So the existential statements are restricted by an extra "and"-clause asserting membership of the set, universal statements are restricted by adding the membership as a precondition in an implication.
So the negation of $\exists x \in A: \phi(x)$ can be $\forall x: (x \notin A) \lor \lnot \phi(x)$ by applying de Morgan to the conjunction and the usual $\forall$ and $\exists$ swap.
The negation of $\forall x \in A: \phi(x)$ is
$$\lnot\left(\forall x: (x \in A) \to \phi(x)\right) \equiv \exists x: (x \in A) \land \lnot \phi(x) :=: \exists x \in A: \lnot \phi(x) $$
the last by abbreviataion again, so the restricted quantifiers behave the same way among each other (with the same $A$) as unrestricted ones.