Axiom of extensionality - why can't we replace $\Rightarrow$ with $\iff$? The axiom of extensionality says that 
$$(\forall(x,y))\big(\forall(z)(z\in x\iff z\in y)\Rightarrow x=y\big)$$ 
Does it not work the other way around?  
$$ (x=y)\Rightarrow\forall z(z\in x\iff z\in y) $$
If two sets are equal and $z$ is in one of them, then - by definiton - it has to be in the other one. 
If it is not true, could you show a counter-example?

The converse of extensionality is a result of the axioms
of a first order logic with equality.  An alternative to
embedding ZF into a first order logic with equality is to
embed it into a first order logic and use the axiom of
extensionality as a definition of equality.  Then the
"converse" would be an iteration of the definition.
