Given are three random variables $X, Y, Z$.
If I now want to calculate e.g. $$ \text{E}\left[Z\cdot \left(Y\middle|X=x\right)\right], $$
does $$ \text{E}\left[Z\cdot \left(Y\middle|X=x\right)\right]= \text{E}\left[Z\cdot Y\middle|X=x\right] $$ hold?
My guess is yes, since - assuming that $X=x$ contains information about $Z$ (i.e. not independent) - it would be kind of strange to have the information that $X=x$ "available" for $Y$, but not for $Z$, right?