Theorem hbntal 45322

Description: A closed form of hbn 2332. hbnt 2331 is another closed form of hbn 2332. (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 2330	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑))
2		axc7 2352	. . . . 5 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑)
3	2	con1i 148	. . . 4 ⊢ (¬ 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4		con3 154	. . . . 5 ⊢ ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
5	4	al2imi 1848	. . . 4 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
6	3, 5	syl5 35	. . 3 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
7	6	alimi 1844	. 2 ⊢ (∀𝑥∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀𝑥 ¬ 𝜑))
8	1, 7	syl 18	1 ⊢ (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥(¬ 𝜑 → ∀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 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-or 862  df-ex 1813  df-nf 1817

This theorem is used by:  hbimpg  45323  hbimpgVD  45672  hbexgVD  45674