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Database CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes) Axiom scheme ax-4 (Quantified Implication) 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