Theorem hbng 30958

Description: A more general form of hbn 2131. (Contributed by Scott Fenton, 13-Dec-2010.)

Hypothesis

Ref	Expression
hbg.1	⊢ (𝜑 → ∀𝑥𝜓)

Assertion

Ref	Expression
hbng	⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)

Proof of Theorem hbng

Step	Hyp	Ref	Expression
1		hbntg 30955	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
2		hbg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	mpg 1715	1 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696

This theorem is referenced by: (None)