Difference between these two statements

Question

1. $\forall x\in S, \forall z\in S,\exists y\in C, (x\neq z) \Rightarrow ...$

2. $\forall x\in S, \forall z\in S, \exists y\in C,...$

Why is there a need for $x \ne z$ in 1. Isn't it already implied that x and z are different?

Thanks.

Answer 1

No. They could be the same.

$\forall x \in \mathbb{Z}\, \forall y \in \mathbb{Z}\,, x-y \neq 0$

is not the same as

$\forall x \in \mathbb{Z}\, \forall y \in \mathbb{Z}\,, (x\neq y) \implies x-y \neq 0$