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Theorem hbntal 39086
Description: A closed form of hbn 2184. hbnt 2182 is another closed form of hbn 2184. (Contributed by Alan Sare, 8-Feb-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbntal (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbntal
StepHypRef Expression
1 hba1 2189 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥(𝜑 → ∀𝑥𝜑))
2 axc7 2170 . . . . 5 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
32con1i 144 . . . 4 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4 con3 149 . . . . 5 ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
54al2imi 1783 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
63, 5syl5 34 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
76alimi 1779 . 2 (∀𝑥𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
81, 7syl 17 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-ex 1745  df-nf 1750
This theorem is referenced by:  hbimpg  39087  hbimpgVD  39454  hbexgVD  39456
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