TITLE: First-order logic: nested quantifiers for same variables
QUESTION [5 upvotes]: Doing some homework, I'm asked to determine if the following formula is satisfiable, valid or neither.
I am confused by the nesting quantifiers for the same variables.
Using sequent calculus, I derive this is not valid, but I think it's valid because the variables are all bounded to the same first quantifier.
$$\exists x\exists y (P (x, y) \rightarrow \forall x\forall y P (x, y))$$
Help would be much appreciated.
Edited to avoid confusion. This is about classical FOL using sequent calculus.

REPLY [2 votes]: $$\exists x\;\exists y\; \left(P (x, y) \rightarrow \forall x\;\forall y \;P (x, y)\right).$$
I wonder if the notion of an "alphabetic variant" can make this clearer:
$$\exists x\;\exists y\; \left(P (x, y) \rightarrow \forall u\;\forall v \;P (u, v)\right).$$
Then you can say that $P(x,y) \rightarrow \forall u\;\forall v \;P (u, v)$ is equivalent to $\forall u\;\forall v \;\left( P(x,y) \rightarrow P (u, v) \right)$,
so the expression above is equivalent to
$$\exists x\;\exists y\; \forall u\;\forall v \;\left( P(x,y) \rightarrow P (u, v) \right).$$