Probability chain rule given some event.

Question

I'm dealing with a probability of the form $\mathbb{P}(\cap_{l \in S} E_l | X)$. Using the conditional probability formula and the probability chain rule would give me a product over the set $\{E_l: l \in S\} \cup \{X\}$. What if I just want to write the product over the set $\{E_l: l \in S\}$?

I am actually trying to understand the following statement, if that helps:

Answer 1

I will try to explain the equation in black, since this is what you are trying to understand. So this is just Bayes applied over and over. Note that: $$\mathbb P(E_1 \cap E_2 \mid \cap_{i\in S_2} A_i) = \mathbb P(E_1 \mid \cap_{i\in S_2} A_i)\mathbb P(E_2 \mid (E_1) \cap (\cap_{i\in S_2} A_i))$$ Now, just use this for your case with $\mathbb P(\cap_{i \in S_1}E_i \mid \cap_{i\in S_2} A_i) $.