How to prove an implication within an if and only if Suppose you need to prove that $A\iff (B\implies C)$.
The two ways to prove this are:

(1a): Suppose $A$ and $B$ are true. Prove that $C$ is true.
(1b): Suppose $B$ and $C$ are true. Prove that $A$ is true.

(2a): Suppose $A$ and $B$ are true. Prove that $C$ is true.
(2B): Suppose $A$ is not true and B is true, prove that $C$ cannot be true.

Are these ways correct? I always get confused what you can assume and what you have to prove when there's multiple implications and such in one statement.
 A: It helps to call $D$ the statement $B\implies C$. One has to prove $A\iff D$. So we need to show that $A$ implies $D$, and $D$ implies $A$. This means again, that, assuming $A$ it must follow $C$ if we assume $B$, and conversely, that whenever $C$ follows from $B$, then $A$ follows. Now check your $4$ statements according to this reasoning.
A: Prove both directions:
Forward: assume A, prove ($B \implies $C). This means "suppose A and B are true. Prove that C is true".
Reverse: Assume ($B \implies $C), prove A. So, you don't  "Suppose B and C are true. Prove that A is true". Instead, "suppose the truth of B implies truth of C, then prove that A is true. $1b$ is incorrect.
Now, reverse can also be interpreted as suppose A is not true, prove ($B \implies $C) is not true by the contrapostive. Which means prove $B$ is true and $C$ is false. So you don't "Suppose A is not true and B is true, prove that C cannot be true." Instead, "Suppose A is not true, prove B is true and prove that C cannot be true." $2b$ is incorrect.
A: Hint 1: There are 6 ways to prove $P\implies Q$:


*

*Assume $P$, then prove $Q$

*Assume $\neg Q$, then prove $\neg P$

*Prove $\neg P$

*Prove $Q$

*Prove $\neg[P \land \neg Q]$

*Prove $\neg P \lor Q$


Hint 2: To prove $P\iff Q$, prove $P\implies Q$, then prove $Q\implies P$
