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Theorem hbntal 38278
Description: A closed form of hbn 2142. hbnt 2140 is another closed form of hbn 2142. (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 2148 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → ∀𝑥𝑥(𝜑 → ∀𝑥𝜑))
2 axc7 2128 . . . . 5 (¬ ∀𝑥 ¬ ∀𝑥𝜑𝜑)
32con1i 144 . . . 4 𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
4 con3 149 . . . . 5 ((𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝜑))
54al2imi 1740 . . . 4 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
63, 5syl5 34 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
76alimi 1736 . 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 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by:  hbimpg  38279  hbimpgVD  38650  hbexgVD  38652
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