Let's take the following random variables $X,Y,Z_1,...,Z_K$, on the same probability space (or associated with the same experiment), where we know that $$ Y = \sum_{k=1}^K\beta_kZ_k$$
Let's say I am interested in computing the conditional variance of $X$ knowing $Y$, similarly as in this question. Applying the law of total variance, we get:
$$ V[X|Y]=V[E[X|Z_1,...,Z_K,Y]|Y]+E[V[X|Z_1,...,Z_K,Y]|Y] $$
In this special case, since when we know $Z_1,...,Z_K$, we also know $Y$, can we reduce the law of total variance to:
$$ V[X|Y]=V[E[X|Z_1,...,Z_K]|Y]+E[V[X|Z_1,...,Z_K]|Y] $$
Any help or suggestions would be much appreciated.