Law of total expectation and conditional expectation We know from law of total expectation that 
$$
\mathbb{E}[\mathbb{E}[Y|X]]=\mathbb{E}[Y]
$$
Does that still work if there is a further condition, i.e. does this equation hold?
$$
\mathbb{E}[\mathbb{E}[Y|X]|Z=z]=\mathbb{E}[Y|Z=z]
$$

No, it does not. A counterexample would be $Y=Z$, $X$ independent of $Y$. Then $
\mathbb{E}[\mathbb{E}[Y|X]] = \mathbb{E}Y$,  $$\mathbb{E}[\mathbb{E}[Y|X]|Z=z] = 
\mathbb{E} Y,$$ $$\mathbb{E}[Y|Z=z] = z.$$
Of course, we need $Y$ to be non-degenerate.
