In our textbook it is stated that
$$E[X \cdot E[Y|Z]] = E[Y \cdot E[X|Z]]$$
The wikipedia page for conditional expectation states it also can be shown that the expressions above are equal to $E[E[X|Z] \cdot E[Y|Z]]$.
Intuitively, I can understand why this is true, but how can I formally prove that the three expressions are equal?
My work:
$$E[X \cdot E[Y|Z]]$$ $$ = E[E[X \cdot E[Y|Z] | Z]]$$ $$ = E[E[X|Z] \cdot E[Y|Z]]$$
Similarly, we can show that $E[Y \cdot E[X|Z]] = E[E[X|Z] \cdot E[Y|Z]]$.
But I'm not sure whether the derivation is correct.