Is this statement true: if $P(A|B \cup C) = P(A)$, i.e. $A$ is independent of $ B \cup C$, then $P(A|B \cap C)=P(A)$, i.e. $A$ is also independent of $B \cap C$
Intuitively it makes much sense to me because if $A$ is independent of a bigger set, then $A$ must be independent of a set which belongs to the bigger set. But I could not able to prove it from definition.
Can some one help me to get the proof?