Metamath Proof Explorer

Theorem r19.28z

Description: Restricted quantifier version of Theorem 19.28 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 r19.3rz.1 ⊢ Ⅎ x φ
Assertion r19.28z ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ φ ∧ ∀ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 r19.3rz.1 ⊢ Ⅎ x φ
2 r19.26 ⊢ ∀ x ∈ A φ ∧ ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
3 1 r19.3rz ⊢ A ≠ ∅ → φ ↔ ∀ x ∈ A φ
4 3 anbi1d ⊢ A ≠ ∅ → φ ∧ ∀ x ∈ A ψ ↔ ∀ x ∈ A φ ∧ ∀ x ∈ A ψ
5 2 4 bitr4id ⊢ A ≠ ∅ → ∀ x ∈ A φ ∧ ψ ↔ φ ∧ ∀ x ∈ A ψ