Proof of semi-graphoid symmetry axiom

I have some trouble understanding why this proof is a proof:

Lemma:

Proof:

Ok, this is just conditional independence, x independent of y given z.

This is rewritten according to the rule of conditional independence P(A|B) = P(A & B) / P(B).

Bottom up the same is done for the right hand side of the lemma, but I don't get how the following and previous step are equal.. Why does this prove anything?

Re: Proof of semi-graphoid symmetry axiom

Hey Lepzed.

Is the random variable X independent to Z?

Re: Proof of semi-graphoid symmetry axiom

The independence relation expresses that in the context of information about Z, information about Y is irrelevant with respect to Y or put differently: X is condtionally independent of Y given Z.