# Conditional expectations and conditional independence

While learning causal models, I came across the below assertions:

to express the conditional independence of Y and X given Z: $$Pr(Y|X, Z) = Pr(Y|Z)$$

we can write:

$$E(Y|X, Z) = E(Y|Z)$$ If X, Y and Z are random variables with multivariate normal distributions.

My question is shouldn't this hold for all distributions?

$$E(Y|X, Z) = \int_{z}\int_{x} y \, dPr(y|x, z) = \int_{x} y \, d Pr(y|x) = E(Y|Z)$$

A bit rusty on the math 12 years after graduation, but what did I miss?

Any place where I can read about a proof using the multinormal property to prove the expectation equation?