Let $X,Y$ be random variables such that $E[Y|X]=X$ and $Z$ independent of $(X,Y)$. Let $W=X+Y+Z$, and $V=E[Y|X,W]$. $V$ is a random variable that depends on $X$ and $W$. What is the conditional probability of $V$ given $(X,Y)$, i.e what is $P(V \in \cdot | X=x,Y=y)$?
My idea: $V$ depends on $X$ and $W$. $X$ is known, and since we know $(X,Y)$, the only randomness left in $W$ is $Z$. But since $Z$ is independent of $(X,Y)$, $W$ becomes independent of $(X,Y)$. So
$$P(V \in \cdot | X=x,Y=y)=P(E[Y|X=x,x+y+Z] \in \cdot | X=x,Y=y)=P(E[Y|X=x,x+y+Z] \in \cdot )=P(E[Y|X=x] \in \cdot )=P(x\in \cdot )$$
Hence the conditional probability is a Dirac at $x$. Is this correct?
This is a very particular case of a more general question I asked here: What is the distribution of $Z=E[X|Y]$ conditional on $X$?
This is not correct. For example, consider the case that $Z=0$ identically. Then $E(Y|X,W)=W-X$.
This is just the simplest example. It is false in general that the conditional distribution of $V$ given $X,Y$ is Dirac at $X$. You can see this also in the case when $|Z|<1$, $|X|<1$ and $Y=X+R$ where $R$ is independent of $X,Z$ and takes values $\pm 10$ with equal probability.