Theorem hbntal 43299

Description: A closed form of hbn 2291. hbnt 2290 is another closed form of hbn 2291. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hbntal	⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbntal

Step	Hyp	Ref	Expression
1		hba1 2289	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑))
2		axc7 2310	. . . . 5 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
3	2	con1i 147	. . . 4 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4		con3 153	. . . . 5 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
5	4	al2imi 1817	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
6	3, 5	syl5 34	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
7	6	alimi 1813	. 2 ⊢ (∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
8	1, 7	syl 17	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786

This theorem is referenced by:  hbimpg  43300  hbimpgVD  43650  hbexgVD  43652