I was just wanting to make sure I had treated everything correctly when negating this statement. I wasn't 100% on how to treat the bracket between the existential quantifiers and whether or not I should change the predicates inside the brackets.

Negate:

$\displaystyle \exists x \in \mathbb{Z}\left(\exists y \in \mathbb{Z}\left(\left((x+1)^2>y^2\right) \rightarrow (x\neq y)\right)\right)$

First of all am I correct in thinking it reads "There exists an x that is an integer such that there exists a y that is an integer such that blah blah implies blah"?

My negation:

$\displaystyle \forall x \in \mathbb{Z} \left(\forall y \in \mathbb{Z} \neg \left(\left((x+1)^2>y^2)\rightarrow (x\neq y)\right)\right)$

$\displaystyle \equiv \forall x \in \mathbb{Z} \left(\forall y \in \mathbb{Z}\neg \left(\neg \left((x+1)^2>y^2)\vee (x\neq y)\right)\right)$

$\displaystyle \equiv \forall x \in \mathbb{Z} \left(\forall y \in \mathbb{Z} \left( \left((x+1)^2>y^2)\wedge\neg (x\neq y)\right)\right)$

$\displaystyle \equiv \forall x \in \mathbb{Z} \left(\forall y \in \mathbb{Z} \left( \left((x+1)^2>y^2)\wedge (x = y)\right)\right)$

How do I determine which statements are true/false. I see it all as one statement is the problem how do I separate them?

Thanks.