I recently came across this identity:

$$E \left[ E \left(y|x,z \right) |x \right] =E \left(y | x \right)$$

I am of course familiar with the simpler version of that rule, namely that $E \left[ E \left(y|x \right) \right]=E \left(y\right) $ but I was not able to find justification for its generalization.

I would be grateful if someone could point me to a not-so-technical reference for that fact or, even better, if someone could lay out a simple proof for this important result.

Thank you.