We can compute probabilities using a chain rule as follows:
P(A &and B &and C) = P(A | B &and C) * P(B | C) * P(C)
If some conditions C1 &and ... &and Cn are independent of other conditions U, we will have:
P(A | C1 &and ... &and Cn &and U) = P(A | C1 &and ... &and Cn)
This allows a conditional probability to be computed more easily from smaller tables using the chain rule.