In Kallenberg's probability text, Lemma 3.11 states what I think is related to the tower law.
If $\xi$ and $\eta$ are independent random variables in measurable spaces $S$ and $T$, and $f: S \times T \to \mathbb{R}$ is measurable with $E(E|f(s,\eta)|)_{s=\xi}<\infty$, then $E f(\xi,\eta)=E(E(f(s,\xi))_{s=\xi}$.
This looks like the tower law (aka "smoothing") $E[E[f(X,Y) \mid X]]=E[f(X,Y)]$, which I have seen in other contexts, but I don't recall there being an independence assumption as there is in the above lemma. In general is there an independence requirement between $X$ and $Y$?