How to prove an implication within an if and only if

Question

Suppose you need to prove that $A\iff (B\implies C)$.

The two ways to prove this are:

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

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

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(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.

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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.

Answer 1

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.

Answer 2

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.

Answer 3

Hint 1: There are 6 ways to prove $P\implies Q$:

1. Assume $P$, then prove $Q$
2. Assume $\neg Q$, then prove $\neg P$
3. Prove $\neg P$
4. Prove $Q$
5. Prove $\neg[P \land \neg Q]$
6. Prove $\neg P \lor Q$

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