Theorem nf5di

Description: Since the converse holds by a1i , this inference shows that we can represent a not-free hypothesis with either F/ x ph (inference form) or ( ph -> F/ x ph ) (deduction form). (Contributed by NM, 17-Aug-2018) (Proof shortened by Wolf Lammen, 10-Jul-2019) Reduce axiom dependencies. (Revised by SN and Sky Winter, 23-Oct-2026)

	Ref	Expression
Hypothesis	nf5di.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜑 )
Assertion	nf5di	⊢ Ⅎ 𝑥 𝜑

Proof

Step	Hyp	Ref	Expression
1		nf5di.1	⊢ ( 𝜑 → Ⅎ 𝑥 𝜑 )
2		hbe1	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 ∃ 𝑥 𝜑 )
3	1	nfrd	⊢ ( 𝜑 → ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 ) )
4	3	com12	⊢ ( ∃ 𝑥 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
5	2 4	eximdh	⊢ ( ∃ 𝑥 𝜑 → ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑥 𝜑 ) )
6	5	pm2.43i	⊢ ( ∃ 𝑥 𝜑 → ∃ 𝑥 ∀ 𝑥 𝜑 )
7		hbe1a	⊢ ( ∃ 𝑥 ∀ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
8	6 7	syl	⊢ ( ∃ 𝑥 𝜑 → ∀ 𝑥 𝜑 )
9	8	nfi	⊢ Ⅎ 𝑥 𝜑