Theorem nf3

Description: Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021)

	Ref	Expression
Assertion	nf3	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nf2	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ¬ ∃ 𝑥 𝜑 ) )
2		alnex	⊢ ( ∀ 𝑥 ¬ 𝜑 ↔ ¬ ∃ 𝑥 𝜑 )
3	2	orbi2i	⊢ ( ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 ¬ 𝜑 ) ↔ ( ∀ 𝑥 𝜑 ∨ ¬ ∃ 𝑥 𝜑 ) )
4	1 3	bitr4i	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ∀ 𝑥 ¬ 𝜑 ) )