Metamath Proof Explorer

Theorem 19.38b

Description: Under a nonfreeness hypothesis, the implication 19.38 can be strengthened to an equivalence. See also 19.38a . (Contributed by BJ, 3-Nov-2021) (Proof shortened by Wolf Lammen, 9-Jul-2022)

Ref Expression
Assertion 19.38b ⊢ Ⅎ x ψ → ∃ x φ → ∀ x ψ ↔ ∀ x φ → ψ

Proof

Step Hyp Ref Expression
1 19.38 ⊢ ∃ x φ → ∀ x ψ → ∀ x φ → ψ
2 exim ⊢ ∀ x φ → ψ → ∃ x φ → ∃ x ψ
3 id ⊢ Ⅎ x ψ → Ⅎ x ψ
4 3 nfrd ⊢ Ⅎ x ψ → ∃ x ψ → ∀ x ψ
5 2 4 syl9r ⊢ Ⅎ x ψ → ∀ x φ → ψ → ∃ x φ → ∀ x ψ
6 1 5 impbid2 ⊢ Ⅎ x ψ → ∃ x φ → ∀ x ψ ↔ ∀ x φ → ψ