Metamath Proof Explorer

Theorem r19.27z

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, 26-Oct-2010)

Ref Expression
Hypothesis r19.27z.1 ⊢ Ⅎ x ψ
Assertion r19.27z ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ψ

Proof

Step Hyp Ref Expression
1 r19.27z.1 ⊢ Ⅎ x ψ
2 1 r19.3rz ⊢ A ≠ ∅ → ψ ↔ ∀ x ∈ A ψ
3 2 anbi2d ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
4 r19.26 ⊢ ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
5 3 4 syl6rbbr ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ψ