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Using the 'do calculus' + DAGS framework for causal inference, is this
If $(Y\perp X)_{G_{\underline{x}}}$ then $P(Y|X=x)=P(Y|\text{do}(X)=x)$
an axiom? Or can it be proven from first principles (and without using any notation from an alternative framework, such as potential outcomes)?
Intuitively, do-calculus is indeed the first axiomatic formulation for causal inference, see do-calculus rules (Axioms) from Stanford Encyclopedia of Philosophy's causal inference entry.
The equality asked can be regarded as the special case of the axioms. This is so-called the back-door criterion $P(Y|do(X),Z) = P(Y|X, Z) $. But it requires more strict conditions than only independence.
Axioms should provide equivalency to potential outcomes, however they have philosophical differences in how they construct solutions of course. One could argue that, DAGs provides more insights in the mechanisms, i.e., explicit causal model.
