Let $X_1,X_2,C,Z$ be random variables.
Is it true that if $(X_1,X_2)\perp\!\!\perp Z\mid C$ then $X_1\perp\!\!\perp Z\mid C$?
I am reading a paper where this seams like it is implied, but cannot prove it. Just using Bayes and algebraically manipulating the equations doesn't directly lead to the result, but intuitively I think it is true: if some component of a vector was dependent on $Z$ then that would compromise the joint independent property for the whole vector...