Is it true that for $Y$ and $Z$ independent, the conditional entropy $H(X\mid Y,Z)$ satisfies $$H(X\mid Y,Z) = H(X\mid Y) + H(X\mid Z)$$ where $H(X\mid Y) = \sum_{y\in\mathcal{Y}}p(y)H(X\mid Y=y)$ (similarly for $H(X\mid Z)$)?
Is this obvious? Can someone point me in the direction of a proof?