I was told that a if and only if b implies b if and only if a. I am not sure I believe this because I can think of many examples where this seems to be false.
The animal is a human if and only if it is a mammal (True). The animal is a mammal if and only if it is a human (False). Therefore a iff b does not imply b iff a.
"The animal is a mammal if and only if it is a human" is clearly false because there are many mammals that are not humans.
"The animal is a human if and only if it is a mammal". Every animal must belong to a taxonomic class. An animal can not be without a class. Every human is a mammal. Therefore the animal is a human if and only if it is a mammal. It does not suffice to say that the animal is a human because it has 10 fingers. We can not ignore or avoid assigning taxonomic class.
Best Answer
"If and only if" is the biconditional connection; a statement of material equivalence.
$A\leftrightarrow B$ is equivalent to $\underbrace{(A\leftarrow B)}_\text{if}\underbrace{\wedge}_\text{and} \underbrace{(A\to B)}_\text{only if}$, and you can see the symmetry there in.
That is that "$A$ if and only if $B$" means "$A$ if $B$, and $A$ only if $B$".
So your proposed counterexample of "An animal is human if and only if it is a mammal" means "An animal is human if it is a mammal, and an animal is human only if it is a mammal." Which is false, and thus actually equivalent to "An animal is a mammal if and only if it is human," which is also false.