Prove that P(A|B,C)=P(A|C,B) after applying the Bayesian updating process

Question

Prove that P(A|B,C)=P(A|C,B) after applying the Bayesian updating process. That is, prove that the order in which the information is presented does not matter.

Answer 1

We know that $$P(A|B,C) = \frac{P(A \cap B \cap C)}{P(B \cap C)}$$

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

$$= P(A|C,B)$$

Answer 2

$$ P(A|B,C)=P(A|B |C)=\frac{P(A,B,C)}{P(B).P(C)}= \frac{P(A,C,B)}{P(C).P(B)}= P(A|C |B)= P(A|C,B)$$