Mathematical explanation for this relationship

I have analyzed some data, and noticed that it fits this relationship quite well:

$$P(A|Bcap C) = P(A|B)frac{P(A|C)}{P(A)}$$

I am wondering what the interpretation of this would be mathematically, e.g. what kind of assumptions about independence or other things you would need to go from the left to right hand side. A priori, I believe events $B$ and $C$ to be independent, so the relationship would reduce to:

$$
frac{P(Acap Bcap C)}{P(B)P(C)} = frac{P(Acap B)P(Acap C)}{P(A)P(B)P(C)}\
P(Acap Bcap C) = frac{P(Acap B)P(Acap C)}{P(A)}\
$$

However, I can’t find a way to take it further than that. Are there other standard statistical identities/properties that help explain this relationship, or is this just its own unique occurrence that doesn’t have further explanation?

Edit: Added $P(A)$ on bottom of right hand side