The first mistake announcement

I hope there won't be many of these posts, but some are unavoidable in a course that goes where no course has gone before. So let's get to it: in class we proved that if $f(X_1,X_2)$ is a function of two independent random variables then
$\mathbb{V}_{X_2}[\mathbb{E}_{X_1}f(X_1,X_2)]\leq \mathbb{E}_{X_1}[\mathbb {V}_{X_2}f(X_1,X_2)]$. We gave a condition for when equality holds above, which was wrong. Finding what is the correct necessary and sufficient condition appears as a question in the first home assignment.