If an event $A$ is independent from $B$ and $C$, is A also independent from $B \cap C$? $A$ and $B$ are independent. $A$ and $C$ are independent. Are $A$ and $B \cap C$ also independent?

$\implies P(A|B)=P(A),P(B|A)=P(B)$
$\implies P(A|C)=P(A),P(C|A)=P(C)$
$P(A|(B∩C))=P(A∩B∩C)/P(B∩C)$
Which after simplification using,
$P(X|Y)=P(X∩Y)/P(Y)$
$P(A|(B∩C)) ≠ P(A)$
