Proof by contradiction on if and only if statements

Question

Suppose I want to prove a general statement like 'A is true if and only if B is true'

If I assumed B is untrue and showed that subsequently A is untrue, which direction am I actually proving? I guess it is the direction going from left to right?

Correct. Proving that $\neg B\implies \neg A$ is equivalent to proving that $A\implies B$. Assuming you finished that step, then what remains to prove the bi-implication is to prove $B\implies A$ or to prove $\neg A\implies \neg B$. — JMoravitz, May 22 '19 at 13:37

score 0 · Accepted Answer · answered May 22 '19 at 13:41

Exactly. If you assumed that $B$ is untrue, and proved that $A$ is then untrue, you have proven the statement $\neg B\implies \neg A$. This statement is logically equivalent to the statement $A\implies B$, which means you proved the direction from left to right.

You still have to prove either $B\implies A$ or its equivalent, $\neg A\implies \neg B$.

Proof by contradiction on if and only if statements

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