Conditional expectation on two variables to conditional expectation in 1 variable

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I am really confused trying to prove the following statement formally although it makes perfect intuitive sense.

I want to show that if $Y$ is conditionally independent on $W$ given $X$, then:

$E[W|X]=E[W|X,Y]$

I know I have to use the Law of Iterated expectations but I am failing to see how to deal with the conditioning.

Thanks!