The identity is the following: 
$E(Y|Z=z) = \int E(Y|Z=z,X=x) P(X=x|Z=z) dx$

I would start by doing:

$E(Y|Z=z) = \int y P(Y=y|Z=z)dy $

and then
$$\int y P(Y=y|Z=z)dy=\int y \int P(Y=y|Z=z,X=x) P(X=x) dx \ dy. \ (1)$$

I know this last step is not right since, for the equality to be true, it should be
$$\int y P(Y=y|Z=z)dy=\int y \int P(Y=y|Z=z,X=x) P(X=x|Z=z) dx \ dy. \ (2)$$

What's the intuition / steep needed to arrive to the equality number 2?

Thanks in advance