Theorem hbng 31714

Description: A more general form of hbn 2146. (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 31711	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
2		hbg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	mpg 1724	1 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)