Bayes theorem for joint probabilities

Question

I want to use bayes theorem to essentially 'swap' the $A$ and $B$ without moving $C$ as in the sense of,

$$ p(A|B,C) \propto p(B|A,C) $$

can this be done and what are the other terms which make this an equality?

Answer 1

I interpret $P(A|B,C)$ to mean $P(A|B\cap C)$ in my answer.

The definition of conditional probability yields $P(A|B\cap C)=\frac{P(A\cap B \cap C)}{P(B\cap C)}$ and $P(B|A\cap C)=\frac{P(A\cap B \cap C)}{P(A\cap C)}$. So $P(B|A\cap C)=\frac{P(A|B\cap C)P(B\cap C)}{P(A\cap C)}$.

Conditional probability formula: https://en.wikipedia.org/wiki/Conditional_probability