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Metamath Proof Explorer

Theorem r19.27zv

Description: Restricted quantifier version of Theorem 19.27 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004)

Ref Expression
Assertion r19.27zv ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ψ

Proof

Step Hyp Ref Expression
1 nfv ⊢ Ⅎ x ψ
2 1 r19.27z ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ψ