Am trying to prove the following generalization of this result.
Proposition. Let $(\Omega,\mathcal F,P)$ be a probability space, and let $X,Y,Z$ be independent random variables taking values in the measurable spaces $(\Omega_X,\mathcal F_X)$,$(\Omega_Y,\mathcal F_Y)$,$(\Omega_Z,\mathcal F_Z)$ respectively. Let $f:(\Omega_X \times\Omega_Y \times\Omega_Z,\mathcal F_X\otimes\mathcal F_Y\otimes\mathcal F_Z)\to\mathbb R$ be measurable. If $f(X,Y,Z)$ is $P$-integrable, then
$$E[f(X,Y,Z)|Y,Z]=E[f(X,y,z)]\circ (Y,Z) \quad P\text{-almost surely}.$$
Proof. Since $X,Y,Z$ are independent, the distribution of $(X,Y,Z)$ on $\Omega_X \times\Omega_Y \times\Omega_Z$ is $P_X\otimes P_Y\otimes P_Z$,where $P_X,P_Y,P_Z$ denote the distributions of $X,Y,Z$ on $\Omega_X,\Omega_Y,\Omega_Z$ respectively. From the $P$-integrability $f(X,Y,Z)$ and the transformation theorem for image measures we get the $P_X\otimes P_Y\otimes P_Z$-integrability of $f(x,y,z)$. Therefore Fubini's theorem implies
$$x\mapsto f(x,y,z) \quad P_X\text{-integrable for } (P_Y\otimes P_Z) \text{-almost every } (y,z), $$
$$(y,z)\mapsto \int f(x,y,z) dP_X \quad (P_Y\otimes P_Z)\text{-integrable}, $$
$$\int f(x,y,z) d(P_X\otimes P_Y\otimes P_Z)=\int \bigg[\int f(x,y,z)dP_X\bigg] d(P_Y\otimes P_Z).$$
In particular $(y,z)\mapsto E[f(X,y,z)]$ is a measurable map defined for $(P_Y\otimes P_Z)$-almost every $(y,z)$. Let $A_Y\in\mathcal F_Y$ and $A_Z\in\mathcal F_Z$. Then another application of Fubini's theorem gives
$$E\big[f(X,Y,Z)1_{Y\in A_Y}1_{Z\in A_Z}\big]=\int f(x,y,z)1_{A_Y}(y)1_{A_Z}(z) d(P_X\otimes P_Y\otimes P_Z)$$
$$=\int \bigg[\int f(x,y,z)1_{A_Y}(y)1_{A_Z}(z) dP_X \bigg] d(P_Y\otimes P_Z)$$ $$=\int 1_{A_Y}(y)1_{A_Z}(z) \bigg[\int f(x,y,z) dP_X \bigg] d(P_Y\otimes P_Z)$$
$$=\int 1_{A_Y}(y)1_{A_Z}(z) E[ f(X,y,z)] d(P_Y\otimes P_Z)$$
$$=E\bigg[1_{Y\in A_Y}1_{Z\in A_Z} E[ f(X,y,z)]\circ (Y,Z)\bigg]$$
Since the collection all sets $\{Y\in A_Y\}\cap\{Z\in A_Z\}$ with $A_Y\in\mathcal F_Y$ and $A_Z\in\mathcal F_Z$ is $\cap$-stable and generates $\sigma(Y,Z)$, an application of the Dynkin's $\pi-\lambda$ theorem shows that in fact
$$E\big[f(X,Y,Z)1_A\big]=E\bigg[1_{A} E[ f(X,y,z)]\circ (Y,Z)\bigg]$$
holds for all $A\in \sigma(Y,Z)$.
Is this correct? Thanks a lot for your help.