I have a notation problem. Let $A$ and $B$ be events such that $P(B)\ne 0$. The by Bayes' theorem, $$ P(A \cap B) = P(A|B)P(B)~~~~~~~~~(1) $$
Similarly, $$ P(A\cap B \cap C) = P(A|B \cap C)P(B\cap C) = P(A|B\cap C)P(B|C)P(C)~.$$
My question is the following:
How would I represent the following: $$ P\left(\cap_{i=1}^{n} A_i\right)$$ using (1).
Added
For $n =4$, I get
$$P\left(\cap_{i=1}^4 A_i\right) = P\left(A_1 |\cap_{i=2}^4 A_i\right)\cdot P\left(A_2 |\cap_{i=3}^4 A_i\right)\cdot P\left(A_3 |\cap_{i=4}^4 A_i\right) \cdot P(A_4)$$