Let $X, Y$ be random variables with values in $\mathcal{X}, \mathcal{Y}$ and $h \colon \mathcal{X} \times \mathcal{Y} \to \mathbb{R}$ be measurable, integrable and whatever.
I want to know if we have a reasonable equality (that means in an almost sure sense) $$ \mathbb{E}\left[h(X, Y) \, \vert \, X = x \right] = \mathbb{E}\left[h(x,Y) \, \vert \, X = x \right]. $$
I convinced myself that this is true in the case where we can disintegrate the joint distribution of $X, Y$.
Is this identity true in a more general situation?