Various theorems, lemmas, and other principles equivalent to the Axiom of Choice (e.g. Zorn's lemma) were argued over until it was established (by Kurt Gödel and Paul Cohen) that the AoC is entirely independent of the ZF axioms, i.e. ZFC and ZF!C are both consistent systems. I think this is the canonical example.

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona