conditional independence of random variables I am trying to proof the following fact for random variables. It's used without proof somewhere. I feel it should be easy, but I cant figure it out.
Let A,B,C and D be four measurable maps from a probability space. We denote independence by $A \bot B$ and conditional independence by $A\bot B |C$.
Now suppose $A \bot B | (C,D)$ and $B\bot D | C$. Then $A \bot B | C$.

For any $a,b\in \mathbb{R}$ (assuming that $A$ and $B$ are real-valued),
\begin{align}
\mathsf{P}(A\le a,B\le b\mid C)&=\mathsf{E}[\mathsf{P}(A\le a, B\le b\mid C,D)\mid C] \\
&=\mathsf{E}[\mathsf{P}(A\le a\mid C,D)\mathsf{P}(B\le b\mid C,D)\mid C] \\
&=\mathsf{E}[\mathsf{P}(A\le a\mid C,D)\mid C]\mathsf{P}(B\le b\mid C) \\
&=\mathsf{P}(A\le a\mid C)\mathsf{P}(B\le b\mid C).
\end{align}
