Theorem hbng

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

	Ref	Expression
Hypothesis	hbg.1	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
Assertion	hbng	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		hbg.1	⊢ ( 𝜑 → ∀ 𝑥 𝜓 )
2		hbntg	⊢ ( ∀ 𝑥 ( 𝜑 → ∀ 𝑥 𝜓 ) → ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 ) )
3	2 1	mpg	⊢ ( ¬ 𝜓 → ∀ 𝑥 ¬ 𝜑 )