Demonstration of a logical biconditional involving 3 statements

Question

I'm working on a demonstration and I stepped into something I've never thought about.
I have to demonstrate something like:

$ A \iff B \iff C $

I usually demonstrate things like this demonstrating a chain of implications, such as:

$ A \implies B $

$ B \implies C $

$ C \implies A $

This time I can't manage to find a way out using a chain of implications.
I happily noticed that I can demonstrate these three:

$ A \land B \implies C $

$ B \land C \implies A $

$ C \land A \implies B $

Basically, I use two statements as hypotesis to demonstrate the third.

Is demonstrating these last three enough to demonstrate the logical biconditional involving A,B and C?

Answer 1

No.

Consider: $A$ is a logical tautology, and $B$ and $C$ are logical contradictions

Then you have:

$A \land B \Rightarrow C$, because $\top \land \bot \Rightarrow \bot$

$B \land C \Rightarrow A$, because $\bot \land \bot \Rightarrow \top$

$C \land A \Rightarrow B$, because $\bot \land \top \Rightarrow \bot$

but clearly you do not have $A \Rightarrow B$