This is not a question about "if a false statement with ZF(C) is considered true, what happens." (The answer for that is trivial, as falsity in classical logic proves everything.) What I am rather asking is what happens if we negate axioms of ZF(C) - in other words, axioms of the new set theory, call it $ZF(C)'''$, are exact negation of each axiom of ZF(C).
When I think of such $ZF'''$ (I am going to drop axiom of choice being negated for now), it has to mirror $ZF$ in some way, so I would expect that all false statements with ZF would now be true statements with $ZF'''$ and true statement with ZF would now seem to be false statement in $ZF'''$. Yet this does not seem to be true also, because I really have not negated laws of natural deduction system in propositional logic, contained in first-order logic.
What really then would be consequences of $ZF'''$?