Theorem hbntal

Description: A closed form of hbn . hbnt is another closed form of hbn . (Contributed by Alan Sare, 8-Feb-2014) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		hba1	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑥 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) )
2		axc7	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → 𝜑 )
3	2	con1i	⊢ ( ¬ 𝜑 → ∀ 𝑥 ¬ ∀ 𝑥 𝜑 )
4		con3	⊢ ( ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ¬ ∀ 𝑥 𝜑 → ¬ 𝜑 ) )
5	4	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ∀ 𝑥 ¬ ∀ 𝑥 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
6	3 5	syl5	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
7	6	alimi	⊢ ( ∀ 𝑥 ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑥 ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )
8	1 7	syl	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜑 ) → ∀ 𝑥 ( ¬ 𝜑 → ∀ 𝑥 ¬ 𝜑 ) )