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?

I have two conditionally independent random variables $A$, $B$ such that $$ P(A,B\mid C) = P(A\mid C)P(B\mid C) . $$ I have to find posterior formula $P(C \mid A,B)$.

My result with a straigthforward application of Bayes rule is $$ P(C \mid A,B) = \frac{P(B\mid C)P(A\mid C)P(A)}{P(A\cap B)} . $$ with few variants (e.g. get an intersection on numerator).

But I can't get the lecturer's solution that is $$ \frac{P(B\mid C)P(C\mid A)}{P(B\mid A)} . $$