Remark

Holds for continuous random variables too

Remark

1. Equality holds iff $I(X;Y|Z)=0 \iff X\to Z\to Y$
2. Can be similarly shown that $I(Y;Z)\ge I(X;Z)$
3. Conditioning Reduces Mutual Information: $$I(X;Y)\ge I(X;Y|Z) \iff X\to Y\to Z$$