Theorem hbng 36372

Description: A more general form of hbn 2330. (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 36369	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
2		hbg.1	. 2 ⊢ (𝜑 → ∀𝑥𝜓)
3	1, 2	mpg 1830	1 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2178  ax-12 2215

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by: (None)