Predicate Logic proof: $\forall x\in S\exists y\in S p(x,y)\to \exists y\in Sp(y,y)$

Question

For any non-empty set $S$ and predicate $p$ defined on $S^2$ prove or disprove $\forall x\in S\exists y\in S p(x,y)\to \exists y\in Sp(y,y)$

I'm trying to prove the above. First of all i need to know whether the statement is true or false.. Here how to change $p(x,y)$ to $p(y,y)$?? pretty confused..

Answer 1

Let $S:=\Bbb Z$ and for all $(x,y)\in S^2$ and let $p(x,y)$ mean $x>y$.

Answer 2

Your statement would be true in some cases, e.g. if $S$ is a singleton.

Your statement would be false, however, if $S=\{ x,y\}$ where $x\neq y$, and

$\forall a,b\in S(P(a,b)\leftrightarrow a\neq b)$.

This doesn't formally disprove your statement, but it does suggest that it is not true in general. Maybe that will suffice for your purposes, but I don't think it is possible to formally prove or disprove your statement as it stands, using ordinary logic and set theory.