Meaning of "$\Leftrightarrow$" I just read (at the beginning of this Wikipedia site) that "$A \Leftrightarrow B$" means "A can be replaced in a logical proof with B". Is this interpretation of the symbol true?
(I have so far understood the symbol for equivalence ($\Leftrightarrow$) to mean that, given $A \Leftrightarrow B$, then if A is true then B is true and vice versa and, if A is false then B is false and vice versa.)

$$\iff \text { means *if and only if* }$$ 
Essentially we have that:
$$A\iff B \to (A \implies B) \text { AND } (B\implies A)$$
"A if and only if B is the same as A if B and B if A"
