How do I simplify the union and intersection of sets. In a solution to a homework problem, it says that $P((A\cup B)\cap C')=P(A\cup B\cup C)-P(C)$. I couldn't figure out how these two are equivalent. I have tried distributing $C$ into $(A\cup B)$ but I do not see where the $-P(C)$ comes from. Could someone explain what is happening here?

Don't forget to condition after shampooing. $$\begin{align} P((A\cup B)\cap C') &= P(C' | A\cup B) \cdot P(A\cup B) \\\ &= [1-P(C|A\cup B)]\cdot P(A\cup B) \\\ &= P(A\cup B) - P(C \cap (A\cup B)) \\\ &= P(A\cup B) - [P(C) + P(A\cup B) -P(C\cup A \cup B)] \\\ &= P(A\cup B \cup C) - P(C). \end{align}$$