Ok! But first I need to know what the exact argument is (i.e. premises and conclusion.)
The equation you gave was an example of the different yet equivalent translation of A if and only if B. Normally, this is not proofable (if that ever was a word. LOL!) because it is not a well formed formula (WFF). I can only suppose that the word "equivalent" is the conclusion indicator because there isn't any other indicator.
Are any of these formulas what you need?
A <--> B / (A & B) v (~A & ~B) (most likely candidate for a proof as it has a premise and conclusion)
(A <--> B) <--> [(A & B) v (~A & ~B)] (Though this one needs a conclusion to form a proof. If you just wanted the argument translated into logic, this is the translation you want. You will not be able to proof it the way it is now, only translate it.)
(BTW, the slash is how I do my conclusion indicator. But there are other ways to signify it, like tridot or turnstyle)