I am currently trying to learn about conducting proofs by using subsets. In my textbook, there is an example on this very matter; however, the seem to do something that is in contradiction with what my professor has said: in the example, they use laws of propositional calculus, yet my professor has claimed that doing this is incorrect, but she herself is guilty of doing this in her own examples (she is a rather incompetent teacher). So, who is at fault, the book or her? Are we allowed to use laws in our proofs by subsets? I have been trying to find information on the internet, but have been unsuccessful.
Edit: As an example problem, "Show that if $A$, $B$, and $C$ are sets, then $(A∩B∩C)^c=A^c∪B^c∪C^c$.