Theorem nf6

Description: An alternate definition of df-nf . (Contributed by Mario Carneiro, 24-Sep-2016)

	Ref	Expression
Assertion	nf6	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nf	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
2		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 𝜑
3	2	19.21	⊢ ( ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
4	1 3	bitr4i	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( ∃ 𝑥 𝜑 → 𝜑 ) )