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Theorem hbntOLD 2130
 Description: Obsolete proof of hbnt 2129 as of 13-Oct-2021. (Contributed by NM, 10-May-1993.) (Proof shortened by Wolf Lammen, 3-Mar-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
hbntOLD (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))

Proof of Theorem hbntOLD
StepHypRef Expression
1 df-ex 1696 . . 3 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
2 19.9ht 2128 . . 3 (∀𝑥(𝜑 → ∀𝑥𝜑) → (∃𝑥𝜑𝜑))
31, 2syl5bir 232 . 2 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ ∀𝑥 ¬ 𝜑𝜑))
43con1d 138 1 (∀𝑥(𝜑 → ∀𝑥𝜑) → (¬ 𝜑 → ∀𝑥 ¬ 𝜑))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1473  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-ex 1696 This theorem is referenced by: (None)
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