# Same name predicates

• October 13th 2011, 08:40 AM
infnty
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:

$\forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y) ) )$

I am confused about the double usage of $\exists y$. Is it allowed to do that twice for the same variable (y) within the same scope? It doesn't make sense to me.

It is allowed, but the two y's have nothing to do with each other. Bound variables can be freely renamed, i.e., $\exists y\,F(y)$ is the same formula as $\exists z\,F(z)$. So $\forall x( A(x) \to \exists y ( B(y) \land \exists y C(x, y)))$ is the same as $\forall x( A(x) \to \exists y ( B(y) \land \exists z C(x, z)))$.