As Brian M. Scott explains, they are logically equivalent.
However, the expression $$A \Leftrightarrow B \Leftrightarrow C$$
is ambiguous. It could mean either of the following.
$(A \Leftrightarrow B) \wedge (B \Leftrightarrow C).$
$(A \Leftrightarrow B) \Leftrightarrow C$
These are not equivalent. So for clarity, if we mean option 1, we should write:
The following are equivalent.
Thus, I would reserve the statement $A \Leftrightarrow B \Leftrightarrow C$ for option 2. It works because, somewhat surprisingly, the $\Leftrightarrow$ operation is not only commutative (obvious!) but surprisingly, it is also associative! That is, TFAE.
- $(A \Leftrightarrow B) \Leftrightarrow C$.
- $A \Leftrightarrow (B \Leftrightarrow C)$.
Note that, although statements in the general form of option 2 don't arise much in the usual way of writing math, nonetheless they're completely fundamental in equational logic.