Like you, I'm puzzled by the form of the conditional "Only if A, then B." It doesn't seem to be idiomatic English. One might say "Only if you go to the party will I go," but one wouldn't say "Only if you go to the party, then I will go." That would be unidiomatic. So I presume that the conditional form you're learning is "Only if A, B" rather than "Only if A, then B." I would interpret "Only if A, B" as stating that A is a necessary condition for B, and therefore implying that B is a sufficient condition for A.
If one wants to say that A is both necessary and sufficient for B, then one can say "If and only if A, B" -- although "A if and only if B" would be a smoother way of saying it. In any case, make sure that your logic teacher really did say "Only if A, then B" and, if so, ask if he/she meant to say "Only if A, B."