Property similar to tower property for conditional expectations

Question

I am wondering if for the random variables $A,B,C$ and $P$ there exists a property which says

$$E[A\mid B = b,C = c] = E[E[A\mid P]\mid B = b,C = c] \text{?}$$

Unfortunately I lack the mathematical knowledge to prove this statement.

Note that this is similar to the tower property for conditional expectations but instead it is conditional on two random variables:

Proof of the tower property for conditional expectations

Best regards

Answer 1

The desired property is: $$E[A\mid B=b,C=c]=E\left[ \: E[A\mid P,B=b,C=c] \mid B=b,C=c\right]$$

The idea is that we are living in a world where $B=b,C=c$, and so all probabilities and expectations should be conditioned on this. The above is true because conditional probability distributions given we live in this world are indeed valid probability distributions. So the tower property works the same way for these. Notice that the above is not the same as your guess, since $E[A\mid P]$ is not the same as $E[A\mid P,B=b,C=c]$, and so your guess was close but not quite correct.