$I(X;Y|Z) = H(Y|Z) - H(Y|X,Z)$
$= \sum_{x,y,z}p(x,y,z) \log \frac{p(x,y|z)}{p(x|z) p(y|z)}$ $= \sum_{x,y,z} p(x,y,z) \log p(x,y|z) - \sum_{x,y,z} p(x,y,z) \log p(x|z) - \sum_{x,y,z} p(x,y,z) \log p(y|z)$
$= - \sum_{y,z} \log p(y|z) \sum_{x} p(x,y,z) + \sum_{x,y,z} p(x,y,z) \log \frac{p(x,y,z)}{p(x|z)}$
$= H(y|z) + \sum_{z} p(z) \sum_{x,y}p(x,y|z) \log \frac{p(y|x,y)p(x|z)}{p(x|z)}$
$= H(Y|Z) - H(Y|X,Z)$
Can you help me check if step 3 and 4 are correct or not for getting the second term ?
Thanks