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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) nfnbi
Description: A variable is nonfree in a proposition if and only if it is so in its
negation. (Contributed by BJ , 6-May-2019) (Proof shortened by Wolf
Lammen , 6-Oct-2024)
Ref
Expression
Assertion
nfnbi ⊢ ( Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑥 ¬ 𝜑 )
Proof
Step
Hyp
Ref
Expression
1
exnal ⊢ ( ∃ 𝑥 ¬ 𝜑 ↔ ¬ ∀ 𝑥 𝜑 )
2
1
imbi1i ⊢ ( ( ∃ 𝑥 ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) ↔ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
3
df-nf ⊢ ( Ⅎ 𝑥 ¬ 𝜑 ↔ ( ∃ 𝑥 ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
4
nf4 ⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
5
2 3 4
3bitr4ri ⊢ ( Ⅎ 𝑥 𝜑 ↔ Ⅎ 𝑥 ¬ 𝜑 )