TITLE: Nonsense from combining two iffs ($\iff$)
QUESTION [17 upvotes]: E and F are independent iff $\frac{p(E \cap F)}{p(F)}=p(E)$. 
Also, E and F are independent iff $p(E | F)=p(E)$
Why do I get nonsense if I combine the two? I'd get: E and F are independent iff $p(E | F) = \frac{p(E \cap F)}{p(F)}$
Why doesn't the iff carry over when I combine the two equations?

REPLY [11 votes]: Here is a simple mathematical example to clearly elucidate the logical error:

An integer $n$ is an even prime iff $n = 2$.
An integer $n$ is an even prime iff $n = 1+1$.
(WRONG!) An integer $n$ is an even prime iff $2 = 1+1$.