I have a question about the conditional expectation with some independence conditions for random variables and $\sigma$-fields.
For a random variable $X$ with $E|X| < \infty $, if $Y_1$ and $ Y_2 $ are random variables such that $\sigma(X,Y_1) $ and $\sigma (Y_2) $ independent, then I want to prove the following. $$ E\left(X | Y_{1}, Y_{2}\right)=E\left(X | Y_{1}\right) \quad \text { a.s. } $$
It seems very intuitive since $Y_2$ information is useless for $X$. But I don't know how to prove it. Can anyone help me?