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Metamath Proof Explorer

Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020) (Proof shortened by Wolf Lammen, 11-Dec-2024)

Ref Expression
Assertion r19.21v ⊢ ∀ x ∈ A φ → ψ ↔ φ → ∀ x ∈ A ψ

Proof

Step Hyp Ref Expression
1 pm2.27 ⊢ φ → φ → ψ → ψ
2 1 ralimdv ⊢ φ → ∀ x ∈ A φ → ψ → ∀ x ∈ A ψ
3 2 com12 ⊢ ∀ x ∈ A φ → ψ → φ → ∀ x ∈ A ψ
4 pm2.21 ⊢ ¬ φ → φ → ψ
5 4 ralrimivw ⊢ ¬ φ → ∀ x ∈ A φ → ψ
6 ax-1 ⊢ ψ → φ → ψ
7 6 ralimi ⊢ ∀ x ∈ A ψ → ∀ x ∈ A φ → ψ
8 5 7 ja ⊢ φ → ∀ x ∈ A ψ → ∀ x ∈ A φ → ψ
9 3 8 impbii ⊢ ∀ x ∈ A φ → ψ ↔ φ → ∀ x ∈ A ψ