Mutually independent Problem

Question

Let $A$ and $B$ and $C$ are mutually independent. Prove $B$ is independent from $(A\cap C)'$

The way is started is

\begin{eqnarray} P(B∩(A∩C)' &=& P(B)P(A\cap C)'\\ &=& P(B \cap (A∩C)' \\ &=& P(B) + P(A \cap C)'-P(B \cup (A \cap C)') \end{eqnarray}

But I keep getting stuck.

Answer 1

\begin{align} P(B \cap (A \cap C)')&= P(B)-P(B \cap A \cap C) \\ &=P(B)-P(B)P(A \cap C)\\ &=P(B)(1-P(A \cap C) ) \\ &=P(B)(P(A \cap C)') \end{align}