This is a silly and basic question however I got myself confused. Suppose $A, B, C$ are r.v, how to expand $p(A|B)$ when $p(C)$, $p(A|C)$ and $p(B|C)$ is known? Does it hold
$$p(A|B) = \int p(A|C)p(B|C)p(C)$$
Would you please show me some derivation steps? Thanks!