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Theorem hbntg 30797
Description: A more general form of hbnt 2024. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
hbntg (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbntg
StepHypRef Expression
1 axc7 1992 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜓𝜓)
21con1i 142 . 2 𝜓 → ∀𝑥 ¬ ∀𝑥𝜓)
3 con3 147 . . 3 ((𝜑 → ∀𝑥𝜓) → (¬ ∀𝑥𝜓 → ¬ 𝜑))
43al2imi 1718 . 2 (∀𝑥(𝜑 → ∀𝑥𝜓) → (∀𝑥 ¬ ∀𝑥𝜓 → ∀𝑥 ¬ 𝜑))
52, 4syl5 33 1 (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wal 1472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983
This theorem depends on definitions:  df-bi 195  df-ex 1695
This theorem is referenced by:  hbimtg  30798  hbng  30800
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