assume $ A $ and $ B $ are independent events. For an event $ C $ such as $ 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 must show that $ P ((A mid C) cap (B mid C)) = P (A mid C) cdot P (B mid C) $

My procedure was like that

$$ P ((A mid C) cap (B mid C)) = P ((A mid C) mid (B mid C)) cdot P (B mid C) $$

$$ = frac {P (AB mid C)} {P (B mid C)} $$

I played until I got 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 could not get from the second part the missing part that is $ P (B mid C) $.

Is my procedure correct? If so, how can I find the second part?