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This result generalizes a version of the conditional Ingleton inequality: if for some distribution $I(X: Y \\mid A) = H(A\\mid X,Y)=0$, then $I(A : B) \\le I(A : B \\mid X) + I(A: B \\mid Y) + I(X : Y)$.\n  We present two applications of our result. 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This result generalizes a version of the conditional Ingleton inequality: if for some distribution $I(X: Y \\mid A) = H(A\\mid X,Y)=0$, then $I(A : B) \\le I(A : B \\mid X) + I(A: B \\mid Y) + I(X : Y)$.\n  We present two applications of our result. 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