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Metamath Proof Explorer

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)

		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	1	nf5rd	⊢ ( 𝜑 → ( 𝜑 → ∀ 𝑥 𝜑 ) )
3	2	pm2.43i	⊢ ( 𝜑 → ∀ 𝑥 𝜑 )
4	3	nf5i	⊢ Ⅎ 𝑥 𝜑