What's the negation of $\exists$ $x\in A$ and $y\in B$ s.t. $C$?

Question

What's the negation of $\exists$ $x\in A$ and $y\in B$ s.t. $C$? Is "$\forall x\in A$ or $\forall y\in B$, $\neg C$" correct?

Answer 1

The statement is written with a confusing "and", which leads to error. It is typically not meaningful to have disjunction or conjunction among quantifiers. The original statement should be interpreted as $$\exists x\in A~ \exists y\in B, C$$ whose negation is $$\forall x \in A~ \forall y\in B, \neg C$$

Answer 2

The negation of $~~~~~~\exists (x,y)\in (A\times B):C~~~~~~$ is $~~~~~~\forall (x,y)\in (A\times B):\lnot C$.

The "and" and "or" in the original question are somewhat misleading, particularly if one does not make the dependence of $C$ on $x$ and $y$ explicit.