Double-negatives in the "not+no" formulation tend to be a special case. The easiest rule (as you realized) would be "If there's a 'no' before the noun (direct o

Double-negatives in the "not+no" formulation tend to be a special case. The easiest rule (as you realized) would be "If there's a 'no' before the noun (direct object?), in an already negated phrase, then ignore the 'no'". It's technically incorrect but common English.