TITLE: Prove that biconditional cannot be expressed in terms of implication alone.
QUESTION [5 upvotes]: Is it sufficient to show that since $A \Leftrightarrow  B$ is equivalent to $(A \Rightarrow B) \land (B \Rightarrow A)$ and as conjunction cannot be expressed using conditional alone, neither can biconditional? I can't think of a convincing argument why this would hold. Maybe with some rearrangement it would still be possible?
I tried to get a contradiction using induction like this answer but could not find any.

REPLY [3 votes]: If you write an expression just involving propositional variables and the connective
$\implies$ focus on the right-most variable, $X$ say. If that takes the value $T$ (true) then
the whole expression takes the value $T$. So the expression cannot be logically equivalent
to $A\iff B$ since if $A$ and $B$ take distinct values the expression must evaluate
to $F$ (false). If $A$ is $X$ then taking $A=T$ and $B=F$ gives a contradiction; likewise
if $B$ is $X$. If neither $A$ nor $B$ is $X$, again take $A=T$ and $B=F$ and $X=T$
to give a contradiction.