Do all "if and only if" statements have proofs that can be made from "if and only if" statements, or are there "if and only if" statements that can only be proved by proving an implication then proving that implication's converse?
My first thought was that showing $A=B$ where $A$ and $B$ are sets would sometimes not allow for deductive arguments of "if and only if" statements, but the more I think about it, the more I think that any "if and only if" statement can be proved directly by an argument of "if and only if" statements.
Does anyone have an example of an "if and only if" statement that can't be proven with "if and only if" statements?
Somewhat related to this, are all proofs by contradiction avoidable? i.e. if a statement is true, can you always prove it directly?
My first idea was the proof that any given irrational number is irrational inevitably uses contradiction, but there's a sense in which proving $\sqrt2$ is irrational by proving that it's not rational is exactly the same as proving it's irrational since, by definition, irrational is the negation of rational. There could also be ways of proving some irrational number is irrational that I don't know of.
Is there some way we can know which proof is the most "direct" proof of a statement? Compare the "directness" so to say?
I suppose that would require a definition of "direct."
