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Metamath Proof Explorer

Theorem hbnt

Description: Closed theorem version of bound-variable hypothesis builder hbn . (Contributed by NM, 10-May-1993) (Proof shortened by Wolf Lammen, 14-Oct-2021)

		Ref	Expression
	Assertion	hbnt	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nf5-1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → Ⅎ 𝑥 𝜑 )
2	1	nfnd	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → Ⅎ 𝑥 ¬ 𝜑 )
3	2	nf5rd	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )