I am trying to better visualize why: $$P(A|B,C) = [P(B|A,C)P(A|C)]/P(B|C)$$ I know we can get this by doing: $$ \,\,\,(1)\, P(A|B,C) = P(A,B|C)/P(B|C) \\ (2)\,P(A,B|C) = P(B|A,C)P(A|C)$$
I can visualize the first equation via Venn Diagram. But I cannot visualize the second equation via Venn Diagram. Any advice on how to think about this problem, particularly the second? Any good (quick and visuals) resources?
Thanks.