In Nested Quantifier questions, can y be equal to x?

Question

For example in the problem:

$\forall x\in R, \exists y \in R: x^3 - y = 0 $

Can y simply be equal to $x^3 $ therefore $x^3 - x^3 $ will always equal 0?

Or is this something which would not be considered correct?

Answer 1

Yes, you can reuse values like this.

Of course, the formula you're looking at might make this uninteresting: for example, if I want to show that a formula of the form $$\exists x\exists y(x\not=y\mbox{ and ... })$$ is true, I need to be looking for a pair of different values for the variables to take on.