That doesn't ring right to me. AC implies a well ordering of the reals. And that every set has a well ordering implies AC. But I doubt that the mere assumption that the reals have a well ordering implies AC.

What proof in particular are you referring to? V=L implies AC, but I don't know a proof of that by merely well ordering formulas.

As far as I understand, the answer is 'no'. A result of Feferman is that no explicitly defined well ordering of the real exists. This is mentioned in Levy's 'Basic Set Theory'.