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In the wiki page of Prenex normal form, there is a rule for conjunction as follows:

(∀xφ)∧ψ is equivalent to ∀x(φ∧ψ) under (mild) additional condition ∃xT or, equivalently, ¬∀x⊥(meaning that at least one individual exists).

I can't understand why we need the existence of a x such that the statement φ holds. If for all x, φ is false, then (∀xφ)∧ψ has no model, so does ∀x(φ∧ψ), and they are still logically equivalent (vacuouslly, though).

Did I omit something important?

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    Indeed there's something of import here to prevent the edge case of vacuous truth when the domain is an empty set and one side (which side?) is always true, that's why the 2nd disjunction case has no such issue since both sides are always true in such a case... Commented Sep 19, 2022 at 4:17

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You don't need the existence of an x such that φ holds. You need the existence of an x at all, i.e. ∃x⊤ means "there exists some x for which True," i.e. that there is any x at all. ⊤ is "top" which is an expression that's always true.

The reason is, if there is no x at all, then ∀x(φ∧ψ) is true. But (∀xφ)∧ψ depends on the truth of ψ. so if there is no x, they are not equivalent.

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