Let $\phi : \mathbb{R} \to \mathbb{R}$. Let $X,Y$ be independent random variable.For each $x \in \mathbb{R}$ define $$Q(x,A):= P[\phi (x,Y) \in A ].$$ show $$P[\phi (X,Y) \in A|X ] = Q(X,A)$$ almost surely.
Now assume $\phi$ is either bounded or non-negative. If $h(x := E(\phi(x, Y))$, then $$E (\phi (X, Y)|X) = h(X),$$ almost surely.