Suppose that $X$,$Y$, and $Y'$ are independent and identically distributed random variables defined on some probability space $(\Omega,\mathcal F,P)$. Assume that $f:\mathbb R^2\to\mathbb R$ is a measurable function so that
$$
f(X,Y)
$$
is a random variable. Consider the conditional expectation
$$
\operatorname E[f(X,Y)\mid\sigma(X)],
$$
where $\sigma(X)$ is the $\sigma$-algebra generated by $X$.

> 
> Is it true that $$\operatorname E[f(X,Y)\mid\sigma(X)]=\operatorname E[f(X,Y')\mid\sigma(X)]$$ almost surely? In other words, does the replacement of $Y$ with an independent and identically distributed copy $Y'$ change the conditional expectation? If it does not, how can we make sure that the conditional expectation remains unchanged?
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> 
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Since the conditional expectation is with respect to the $\sigma$-algebra generated only by $X$, it seems that it should indeed be the case. Roughly speaking, the conditional expectation only depends on $X$ and we change $Y$ with an independent and identically distributed copy $Y'$. But I am trying to come up with some sort of a more rigorous argument.

Any help is much appreciated!