Mutual information data-processing inequality

For any bipartite quantum state $\rho_{AB}$ and local quantum channels $\mathcal{N}_{A \to A'}$ and $\mathcal{N}_{B \to B'}$, let $\omega_{A'B'} := (\mathcal{N}_{A \to A'} \otimes \mathcal{N}_{B \to B'})(\rho_{AB})$. Then, the mutual information satisfies $I(A; B)_\rho \ge I(A'; B')_\omega$.