Is this a tautological argument?

Question

Assume $A$ is true, and that $A$ implies $B$. If I then can prove that if $A$ implies $B$, then $B$ implies $A$, have I then proved that $A$ is true (without the need to assume that $A$ is true in the first place)?

Answer 1

From $$A \implies B \\ (A \implies B) \implies (B \implies A)$$ the only possible conclusion is that $$ A \iff B$$ So, $A$ and $B$ are either both true or both false.