Can a conditional probability turn into joint probability?

Question

I am studying stochastic processes on Gardiner's book. On page 43 it says: "by definition of the conditional..."

Let me call $A=(x_1,t_1), B=(x_2,t_2), C=(x_3,t_3)$. What this is saying is that $P(A\cap B|C)=P(A|B\cap C)P(B|C)$. The only way I am able to explain this in terms of the definition is by using $P(D\cap E)=P(D|E)P(E)$ in the following way: $$P(A\cap B|C)=P(A\cap (B|C))=P(A|B|C)P(B|C)=P(A|B\cap C)P(B|C)$$ But is it correct to write $P(A|B|C)=P(A|B\cap C)$?

Answer 1

$$\begin{array}{l} P\left( {A \cap B \cap C} \right) = \frac{{P\left( {\left( {A \cap B} \right) \cap C} \right)}}{{P\left( C \right)}}P\left( C \right) = P\left( {\left( {A \cap B} \right)|C} \right)P\left( C \right) \\ P\left( {A \cap B \cap C} \right) = \frac{{P\left( {A \cap \left( {B \cap C} \right)} \right)}}{{P\left( {B \cap C} \right)}}\frac{{P\left( {B \cap C} \right)}}{{P\left( C \right)}}P\left( C \right) = P\left( {A|\left( {B \cap C} \right)} \right)P\left( {B|C} \right)P\left( C \right) \\ \end{array}$$