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

Lemma:

$\displaystyle I_p(X, Z, Y) \iff I_p(Y, Z, X)$

Proof:

$\displaystyle I_p(X, Z, Y) \iff P(C_x | C_y \land C_z) = P(C_x | C_z)$

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

$\displaystyle \iff \frac{P(C_x \land C_y \land C_z)}{P(C_y \land C_z)} = \frac{P(C_x \land C_z)}{P(C_z)}$

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?

$\displaystyle \iff \frac{P(C_x \land C_y \land C_z)}{P(C_x \land C_z)} = \frac{P(C_y \land C_z)}{P(C_z)}$

$\displaystyle \iff P(C_y | C_x \land C_z) = P(C_y | C_z)$

$\displaystyle \iff I_p(Y, Z, X)$