Is it is permissible to have a double conditional in probability? I've searched google and this site but couldn't find anything like the following:
$P(A | B | C)=?$
If it is legit, would it be equivalent to the following?
$$ \begin{align*} P(A | B | C) &= \frac{P(A \cap B \cap C)}{P(B \cap C)} \\ &= \frac{P(A|B \cap C)P(B \cap C)}{P(B \cap C)} \\ &= P( A|B \cap C) \end{align*} $$
It is permissible to condition on multiple events with a single conditional, and it is indeed the case that:
$$\mathbb{P}(A | B \cap C) = \frac{\mathbb{P}(A \cap B \cap C)}{P(B \cap C)}.$$
Given that this quantity is already well-defined with a single conditional, there does not appear to be any point to defining probability statements with more than one conditional in this exact same way. It is of course open to you to define statements of this kind and then show that they have some usefulness that is not already encapsulated in the existing theory, but if you were to do this you would need to give the double conditional a meaning that is distinct from conditioning on multiple events.