
Same name predicates
Hey all,
I'm doing some homework and one of the exercises got me confused. In particular, it states the following formula:
$\displaystyle $\forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y) ) )$
I am confused about the double usage of $\displaystyle $ \exists y $$. Is it allowed to do that twice for the same variable (y) within the same scope? It doesn't make sense to me.
Thanks in advance!

Re: Same name predicates
It is allowed, but the two y's have nothing to do with each other. Bound variables can be freely renamed, i.e., $\displaystyle \exists y\,F(y)$ is the same formula as $\displaystyle \exists z\,F(z)$. So $\displaystyle \forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y)))$ is the same as $\displaystyle \forall x( A(x) \to \exists y ( B(y) \land \exists z C(x, z)))$.
While having nested bound variables with the same name is legal, unless one wants to make some point about scoping, it is a bad style to use the same name twice, and it is often a typo.