Theorem nf5

Description: Alternate definition of df-nf . (Contributed by Mario Carneiro, 11-Aug-2016) df-nf changed. (Revised by Wolf Lammen, 11-Sep-2021)

	Ref	Expression
Assertion	nf5	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		df-nf	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
2		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 𝜑
3	2	19.23	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
4	1 3	bitr4i	⊢ ( Ⅎ 𝑥 𝜑 ↔ ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )