Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. If $A_1,\dots,A_n\in\mathcal F$ are mutually independent, then for every choice $B_i\in\{A_i,A_i^c\}$, the events $B_1,\dots,B_n$ are mutually independent.
Probability & StatisticsProbability Theory
Discussion
If two events are independent, replacing either event by its complement preserves independence of the resulting pair.