Theorem nfi

Description: Deduce that x is not free in ph from the definition. (Contributed by Wolf Lammen, 15-Sep-2021)

	Ref	Expression
Hypothesis	nfi.1	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
Assertion	nfi	⊢ Ⅎ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		nfi.1	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
2		df-nf	⊢ ( Ⅎ 𝑥 𝜑 ↔ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
3	1 2	mpbir	⊢ Ⅎ 𝑥 𝜑