For a compound event $E_1E_2$,
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$|$E_1$)
such that if $E_1$ and $E_2$ are independent events, we can say that:
Pr($E_1E_2$) = Pr($E_1$)Pr($E_2$)
Is there an analogous formula that can be derived for the conditional probability: Pr($E_1E_2$|$E_3$)?
I would like to be able to reduce Pr($E_1E_2$|$E_3$) to terms that do not involve all three of the events $E_1$, $E_2$, $E_3$, given the assumption that only $E_1$ and $E_3$ are independent.
If this cannot be done, then what assumptions would be necessary in order to be able to do it?