Theorem nf2

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

	Ref	Expression
Assertion	nf2	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ¬ ∃ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nf	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
2		imor	⊢ ( ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( ¬ ∃ 𝑥 𝜑 ∨ ∀ 𝑥 𝜑 ) )
3		orcom	⊢ ( ( ¬ ∃ 𝑥 𝜑 ∨ ∀ 𝑥 𝜑 ) ↔ ( ∀ 𝑥 𝜑 ∨ ¬ ∃ 𝑥 𝜑 ) )
4	1 2 3	3bitri	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∀ 𝑥 𝜑 ∨ ¬ ∃ 𝑥 𝜑 ) )