I've been working under the assumption that
$$ (1)\qquad \mathbf{E}[Y|X] = \mathbf{E}[\mathbf{E}(Y|X)|X] $$
in order to be able to conclude that $ \mathbf{E}[Y|X] = \mathbf{E}[\mathbf{E}(Y|X,Z)|X] $
However, I want to know that what I am assuming (1) is actually correct, but I don't seem to come up with an idea to prove it besides just intuition. My thought process is: the expected value of the expectation of a variable (Y) given another (X), given the same variable (X) does not change the fact that the only given variable is X. Thanks in advance, I've been working on it for hours and I know there must be something obvious I'm missing, but I'm too frustrated to notice at this point.