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Metamath Proof Explorer

Theorem 19.38a

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

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

Proof

Step Hyp Ref Expression
1 19.38 ⊢ ∃ x φ → ∀ x ψ → ∀ x φ → ψ
2 id ⊢ Ⅎ x φ → Ⅎ x φ
3 2 nfrd ⊢ Ⅎ x φ → ∃ x φ → ∀ x φ
4 alim ⊢ ∀ x φ → ψ → ∀ x φ → ∀ x ψ
5 3 4 syl9 ⊢ Ⅎ x φ → ∀ x φ → ψ → ∃ x φ → ∀ x ψ
6 1 5 impbid2 ⊢ Ⅎ x φ → ∃ x φ → ∀ x ψ ↔ ∀ x φ → ψ