Metamath Proof Explorer

Theorem 19.23t

Description: Closed form of Theorem 19.23 of Margaris p. 90. See 19.23 . (Contributed by NM, 7-Nov-2005) (Proof shortened by Wolf Lammen, 13-Aug-2020) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021) (Proof shortened by BJ, 8-Oct-2022)

Ref Expression
Assertion 19.23t ⊢ Ⅎ x ψ → ∀ x φ → ψ ↔ ∃ x φ → ψ

Proof

Step Hyp Ref Expression
1 19.38b ⊢ Ⅎ x ψ → ∃ x φ → ∀ x ψ ↔ ∀ x φ → ψ
2 19.3t ⊢ Ⅎ x ψ → ∀ x ψ ↔ ψ
3 2 imbi2d ⊢ Ⅎ x ψ → ∃ x φ → ∀ x ψ ↔ ∃ x φ → ψ
4 1 3 bitr3d ⊢ Ⅎ x ψ → ∀ x φ → ψ ↔ ∃ x φ → ψ