I am having trouble figuring out the proof for the following question, maybe it's because I am not familiar with the regular conditional distributions.
Let $X, X', Y,Y'$ be real random variables defined on a common probability space, and denote $\sigma(X), \sigma(X')$ the $\sigma$ algebras generated by $X, X'$.
Prove that if the two dimensional random vectors $(X, Y)$ and $(X', Y')$ have same joint distribution, then the random variables $$\mathbb{E}(Y|\sigma(X)) \quad \text{ and } \quad \mathbb{E}(Y'|\sigma(X'))$$ have same distribution.
I tried to use different properties of conditional distribution but have run into trouble or into a mess, then I considered Radon Nikodym theorem but could not find useful argument.