Predicting labels which are not independent of one another
In this case the easiest way is to use a simple single-label classifier with three labels:


A0B1
A1B1
A0B0


As long as the number of combinations is reasonable, that's what will give the best results while satisfying the constraints.

By the way "If label B is false, then label A is false" is logically implied by "if label A is true, then label B is also true". (contrapositive).