Applying Bayes' Theorem in Deduction

Bayes theorem for 3 events

A version of Bayes' theorem for 3 events results from the addition of a third event $C$, with $P(C)>0$, on which all probabilities are conditioned:

$P(A\vert B\cap C)={\frac {P(B\vert A\cap C)\,P(A\vert C)}{P(B\vert C)}}$

Derivation

Using the chain rule

$P(A\cap B\cap C)=P(A\vert B\cap C)\,P(B\vert C)\,P(C)$

And, on the other hand

$P(A\cap B\cap C)=P(B\cap A\cap C)=P(B\vert A\cap C)\,P(A\vert C)\,P(C)$

The desired result is obtained by identifying both expressions and solving for $P(A\vert B\cap C)$.