Suppose $A $ and $B$ are independent events. For an event $C $ such that $P(C) > 0$ , prove that the event of $A$ given $C $ is independent of the event of $B$ given $C $
We have A and B are independent so $P (AB) = P (A) \cdot P (B) $
We need to show that $P ((A\mid C)\cap (B\mid C)) = P (A\mid C)\cdot P (B\mid C)$
My procedure was like this $$P ((A\mid C)\cap (B\mid C)) = P (AB \mid C) $$
I played arount to get this $$ \frac {P (AC)}{P (C)} \cdot \frac {P (B \mid AC)}{P (B \mid C)} $$ Now the first part gives us $P (A \mid C) $ . I couldn't get from the second part the missing part which is $P (B \mid C) $.
Is my procedure correct? If so, how can I find the second part?