Theorem hbng 33053

Description: A more general form of hbn 2303. (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 33050	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
2		hbg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	mpg 1798	1 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by: (None)