Let $X,Y,Z$ three mutually independent random variables. Let $f$ an arbitrary function.
Do we have the following ? $$ \mathbb{E}\left( f\left( X,Y\right) \big| X,Y,Z \right) = \mathbb{E}\left( f\left( X,Y\right) \big| X,Y \right) $$
The tower property of conditional expectation does not seem to help. Also, does the equality fail if we drop the independence assumption ?
As you depend on something that is not used in the function itself and is independent of all variables, $Z$ can be dropped. If these variables are not independent the equality is false in general.