Suppose unconfoundedness holds for a set of potential outcomes $(Y(1),Y(0))$ and treatment $Z$, conditional on a set of covariates, $X$ such that:
$$ (Y(1),Y(0)) \perp Z \mid X $$
Then, is it necessarily the case that conditioning on an extra set of covariates $X'$, still causes the relationship to hold? In other words, does the equation above imply:
$$ (Y(1),Y(0)) \perp Z \mid X, X' $$
It appears that if we were working with a weaker form of unconfoundedness (focusing on just $Z=1$), where we assume the relation holds only on the expectation:
$$ E(Y(1) \mid Z=1, X) = E(Y(1) \mid X) $$
then this should hold for an extended set of covariates, where:
\begin{align} E(Y(1) \mid Z=1, X, X') &= E(Y(1) \mid X,X') \\ \end{align}
My reasoning is that if $E(Y(1) \mid Z=1, X) = E(Y(1) \mid X)$, then it implies that conditioning at levels of $X$, it is sufficient for us to compare outcomes as coming from a randomized study. If we were to condition on a deeper level, at a level of $X$ and a level of $X'$, then outcomes can still be compared, since it was already established that comparison could occur at $X$, so anything deeper should be as well.
Is the above argument sound reasoning? thank you!