Interesting problem. An obvious idea is to try proving this by induction on formulas. However, as often happens, one needs a stronger induction statement because in a formula A <-> B, A and B separately don't have to be tautologies or contain even number of occurrences of each variable even if the whole formula does this. So the challenge is to come up with a generalization of the claim that will let induction go through.
I'll denote truth values by T and F. Let is call a formula A(p) containing a variable p stable with respect to p if for all values of other variables, A(T) and A(F) have the same truth values, i.e., A(T) <-> A(F) is a tautology. Similarly, A(p) is called unstable w.r.t. p if for all values of other variables, A(T) and A(F) have the opposite truth values, i.e., A(T) <-> A(F) is a contradiction. Note that if A is not stable w.r.t. p, it does not necessarily follow that it is unstable w.r.t. p.
Lemma 1. Let a formula A(p) be built from variables, truth values, equivalences and negations only. Then A is stable w.r.t. p if the number of occurrences of p is even, and A is unstable w.r.t. p if the number of occurrences of p is odd.
Proved by induction on A.
Corollary. If a formula as above contains even number of occurrences of every variable, then it is either a tautology or a contradiction.
Lemma 2. A formula built from T, equivalences and negations only is true iff it contains an even number of negations.
Proved by induction.
From this the claim follows.