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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?
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.
