If the proof's validity depends on the intended interpretation of quantifiers over all subsets, then the semantic reading is doing essential work that the syntactic reading cannot replicate.

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Subsets(as used in set theory and logic): Smaller collections taken from a larger collection; for example, all red cars are a subset of all cars.
doing essential work(as used in philosophical analysis): Playing a necessary role that cannot be replaced or skipped—something that is crucial to making something else true or possible.
proof validity(as used in logic): Whether an argument's reasoning is logically correct and follows proper rules of reasoning, so that if the starting claims are true, the conclusion must be true.
quantifiers(the logical form Russell said descriptions should take): Words like 'all,' 'some,' and 'none' that express how many things we're talking about; Russell argued that phrases like 'the king of France' should be understood using these quantity-words rather than as simple names.

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