I have a problem that I can't work out 

I've two conditional independent A,B  such as 

$P(A,B|C) = P(A|C)P(B|C)$

Now I've to find posterior formula for:

$P(C | A,B)$, now what I got was pretty straigthforward application of bayes:

$\frac{P(B|C)P(A|C)P(A)}{P(A\cap B)}$

With few variants (e.g. get an intersection on numerator)

 but I can't get the lecturer solution that is:

$\frac{P(B|C)P(C|A)}{P(B|A)}$

Any help?