yes, "if" statments in proofs are the "p->q" forms from logic, if you have p you also have q (therefore if no q then no p). If you imagine the set of all possible logical propositions as nodes in a graph, p->q tells you that q is reachable from p. If you prove that q is not reachable from whatever axioms you have (i.e. false), then p is false too by necessity, because otherwise q would be reachable and we know that's not the case. But if all you have is that q is reachable, that's not enough to assert that p is also reachable. Because you have no reason to believe that the road from p to q is the only one, it might very well be that p is false but q is reachable from another statment s or something.

"if and only if" is "p->q and q->p" (also equivalent to "p->q and not(p)->not(q), the more intuitive sense of the phrase), it basically establishes equivalence: any proof that requires p also requires q and vice versa, any proof that guarantees p also guarantees q and vice versa.