I believe I may have solved this properly, but I have some nagging doubts about my process. Here is the question.

Show whether the set equality

is true, using set identities.

My process has led me to believe it is false, but I can't shake the feeling that I made a mistake somewhere. Here is my work.

De Morgan's Law

Commutative Law

Distributive Laws (Twice) (this is where I think I screwed up)

Complement Laws (multiple times)

Identity Laws

I do not know whether or not my application of the Distributive Laws in that case was correct. Could someone please check this?