Theorem nfnf1

Description: The setvar x is not free in F/ x ph . (Contributed by Mario Carneiro, 11-Aug-2016) Remove dependency on ax-12 . (Revised by Wolf Lammen, 12-Oct-2021)

	Ref	Expression
Assertion	nfnf1	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		df-nf	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
2		nfe1	⊢ Ⅎ 𝑥 ∃ 𝑥 𝜑
3		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 𝜑
4	2 3	nfim	⊢ Ⅎ 𝑥 ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
5	1 4	nfxfr	⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝜑