Expected value of Normal CDF at normal value

Question

I am working on the problem below:

The solution contains the following line that I don't understand:

Why is the expectation given X?

Answer 1

Just like when two events $A$ and $B$ are independent, we have $P(A\mid B)=P(A)$, when two random variables $X$ and $Y$ are independent, it is true that $E[\,Y\mid X\,]=E[Y]$. The author applies this trick with $Y=\Bbb I(Z\le X)$. This explains why they are allowed to write what they wrote; in order to understand why what they wrote is useful to the rest of the proof, we would need to see the rest of the proof.