Theorem hbntg

Description: A more general form of hbnt . (Contributed by Scott Fenton, 13-Dec-2010)

	Ref	Expression
Assertion	hbntg	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		axc7	⊢ ( ¬ ∀ 𝑥 ¬ ∀ 𝑥 𝜓 → 𝜓 )
2	1	con1i	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ ∀ 𝑥 𝜓 )
3		con3	⊢ ( ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ¬ ∀ 𝑥 𝜓 → ¬ 𝜑 ) )
4	3	al2imi	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ∀ 𝑥 ¬ ∀ 𝑥 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )
5	2 4	syl5	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )