Suppose some condition A is identical to some condition B; to be concise, let's write A=B. It seems obvious, then, that A is necessary and sufficient for B; or more compactly, A<=>B.
On the other hand, that implication's converse (i.e. that A<=>B implies A=B) seems like it isn't right, because we can easily come up with counter-examples. Take my mother, for example; she is always saying, "eating spinach everyday is a necessary and sufficient condition for becoming strong." In other words, she claims that you will become strong if, and only if, you eat spinach everyday. Surely it does not follow that becoming strong is identical to eating spinach...right?
Now I am tempted to consider the question in the context of sets. Suppose you want to prove that two sets S and T are equal. Then it is sufficient to prove that membership in one follows from membership in the other, and vice versa. I.e. x is an element of S <=> x is an element of T. So it appears that the "=" relation follows from "<=>" relation.
Unfortunately, I do not understand where I am going wrong. If it is easy to see why I am confused, could you help clarify this for me?