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Theorem tbw-negdf 1623
Description: The definition of negation, in terms of and . (Contributed by Anthony Hart, 15-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbw-negdf (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)

Proof of Theorem tbw-negdf
StepHypRef Expression
1 pm2.21 120 . . 3 𝜑 → (𝜑 → ⊥))
2 ax-1 6 . . . . 5 𝜑 → ((𝜑 → ⊥) → ¬ 𝜑))
3 falim 1497 . . . . 5 (⊥ → ((𝜑 → ⊥) → ¬ 𝜑))
42, 3ja 173 . . . 4 ((𝜑 → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑))
54pm2.43i 52 . . 3 ((𝜑 → ⊥) → ¬ 𝜑)
61, 5impbii 199 . 2 𝜑 ↔ (𝜑 → ⊥))
7 tbw-bijust 1622 . 2 ((¬ 𝜑 ↔ (𝜑 → ⊥)) ↔ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥))
86, 7mpbi 220 1 (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wfal 1487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 197  df-tru 1485  df-fal 1488
This theorem is referenced by:  re1luk2  1635  re1luk3  1636
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