What do you mean by AC is provably false in intuitionistic logic? Intuitionistic logic + AC has as models ordinary models of classical logic since intuitionistic logic is a refinement of classical logic.

Do you mean [Diaconescu's theorem](https://en.wikipedia.org/wiki/Diaconescu's_theorem) which shows that AC implies excluded middle?

Maybe you mean there are models of intuitionistic logic where AC (and EM) is false?