Prove that the complement of A and B are independent events?

first prove *: if A intersect B is independent, then A intersect B' is independent. (this is on wiki answers)

P(A' intersect B') = P(B')P(A'|B') by definition

= P(B')[1-P(A|B')] since 1 = P(A) + P(A')

= P(B')[1 - P(A)] from the first proof *

= P(B')P(A') since 1 = P(A) + P(A')

conclude with P(A' intersect B') = P(B')P(A') and is therefore independent by definition.

*note* i am a student in my first semester of probability so this may be incorrect, but i used the first proof* so i figured i would proof this one to kinda "give back".