Let $X$ and $Y$ be two random variables (say real numbers, or vectors in some vector space). It seems to me that the following is true:
E [ X | E [ X | Y ] ] = E [ X | Y]
Note that E [ X | Y ] is a random variable in it's own right. Also note that equality here is point-wise, for every point in the sample space of the joint distribution on on $(X,Y)$. My question, assuming I'm not missing something and the above is true, is whether this law has a name, or is written down / proved somewhere.