Context : find a counter example to the statement $$∀x∈C(∃y∈C(y∈A)⇒¬(x∈B))$$
Suppose I know that there is some x such that : x belongs to set C but NOT to set B.
Am I allowed to derive from this that :
for all y (y does not belong to C OR y does not belong to A OR x does not belong to B)
It seems to me that " x does not belong to B" being true, that makes the whole disjunction true for any variable ( even if I have no information on this variable coming from elsewhere).
Is there an official rule that makes this derivation legal in predicate logic natural deduction?
Indeed, when $C$ and $B$ are disjoint sets, then the statement would be true. So that can not serve as a counterexample.
Your approach is no good. You should seek an interpretation of the sets which makes the statement false.
Now, recall that an implication is false only when the antecedent is true but the consequent is false.
So you want $A,B,C$ to be such that the negation of the statement, ie $\exists x\in C~((\exists y\in C~(y\in A))\wedge(x\in B))$, is true.