Let $X_1$ and $X_2$ be two real valued random variables such that we have the conditional density of $X_1$ given $X_2$, i.e. $$\mathbb P(X_1\in M\mid X_2) = \int_M \phi(x_1\mid X_2)dx_1$$Also, let $h$ be a function such that $h(X_2)$ is integrable. For some $u_1>0,u_2\ne 0$, what further assumptions or explanations do we need to do the following step:
$$\mathbb E[h(X_2)1_{\{u_1X_1+u_2X_2\le z\}}] = \mathbb E \left[h(X_2) \mathbb P\left(X_1\le \frac{z-u_2X_2}{u_1}\mid X_2\right)\right]$$
i.e. how could we justify this step?
None. For every bounded random variable $Y$, $$ E(h(X_2)Y\mid X_2)=h(X_2)\,E(Y\mid X_2), $$ hence $$ E(h(X_2)Y)=E(h(X_2)\,E(Y\mid X_2)). $$ Apply this to $$ Y=\mathbf 1_{u_1X_1+u_2X_2\leqslant z}. $$