How to prove $(A\Longrightarrow B)\iff C$?

Question

Can someone teach me how to prove the statement $(A\Rightarrow B)\Leftrightarrow C$?

My idea is:

1. For $\Rightarrow $ part, we assume that A and B are true then show C is true
2. For $\Leftarrow $ part, suppose that C and A are true then show B is true

Am I right? Thanks for your help!

Answer 1

Your approach for 2 is good, but not the one for 1, because there are other ways to make $A \rightarrow B$ true other than both $A$ and $B$ being true (e.g. if both are false, or when $A$ is false and $B$ is true)

For 1, you could try to prove the contrapositive: If $C$ if false, then $A \rightarrow B$ is false. Since $A \rightarrow B$ is false only when $A$ is true and $B$ is false, this amounts to showing that if $C$ is false, then $A$ is true and $B$ is false.