Reduction of bridge positions along bridge disks

We consider a reduction of a knot in a bridge position along two disjoint bridge disks D1 and D2. Suppose that a reduction of the knot along any subset of {D1,D2} results in a bridge position, where we get two distinct new bridges after the reduction along D1∪D2. We show that there are two disjoint bridge disks E1 and E2 such that (D1,E1) and (D2,E2) are cancelling pairs and (D1∪E1)∩(D2∪E2)=∅.