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?