Proving an if and only if statement using two contrapositive statements

Question

I'm currently working on proving an iff statement and was wondering is it allowed for me to prove the two statements required to prove an iff statement using two contrapositive statements.

(for example, I'd prove if A then B using if ~B then ~A) and similarly for the second statement

Thank you!

Answer 1

Yup, totally ok!

To be clear, in order to prove $A \Leftrightarrow B$, you need to prove 2 things:

Part 1. Either prove $A \Rightarrow B$, or prove its contrapositive $\neg B \Rightarrow \neg A$

Part 2. Either prove $B \Rightarrow A$, or prove its contrapositive $\neg A \Rightarrow \neg B$

So, this gives 4 possible ways of proving $A \Leftrightarrow B$:

1. Prove $A \Rightarrow B$ and $B \Rightarrow A$

2. Prove $A \Rightarrow B$ and $\neg A \Rightarrow \neg B$

3. Prove $\neg B \Rightarrow \neg A$ and $B \Rightarrow A$

4. Prove $\neg A \Rightarrow \neg B$ and $\neg B \Rightarrow \neg A$

So note (and this is what @Ahusain warns against in the Comments) what does not work, is to prove $A \Rightarrow B$ and $\neg B \Rightarrow \neg A$ ... because then you have proved the same half of the proof (Part 1) twice! Likewise, proving $B \Rightarrow A$ and $\neg A \Rightarrow \neg B$ is proving the same thing (Part 2) twice. But this is a very common mistake. No joke. If I had a nickel for every time I have seen people do that ....

Answer 2

In order to prove $A \iff B$ you need to prove (a) $A \implies B$ and (b) $B \implies A$. How you prove these doesn't matter (as long as you don't accidentally use any kind of circular logic).

So proving $A \implies B$ directly and then separately proving $\lnot A \implies \lnot B$ to prove $B \implies A$ is 100% valid.