Conditional probability times conditional probability

Question

I am not sure if the Bayes rule can be used under the following conditions, can anyone help me to determine if the equation is right? $$ P(A\mid B) \cdot P(B\mid C) = P(A,B\mid C) $$ $$ P(y\mid A, B) \cdot P(A\mid C) = P(y, A\mid B,C) $$

Answer 1

Since $$ P(A\mid B)=\frac{P(A\cap B)}{P(B)} $$ and $$ P(B\mid C)=\frac{P(B\cap C)}{P(C)} $$ and $$ P(A,B\mid C)=\frac{P(A\cap B\cap C)}{P(C)} $$ The product of the first two is equal to the third precisely when $$ \frac{P(A\cap B\cap C)}{P(B\cap C)}=\frac{P(A\cap B)}{P(B)} $$ which says $P(A\mid B,C)=P(A\mid B)$