Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tbw-negdf Structured version   Visualization version   GIF version

Theorem tbw-negdf 1700
 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 pm2.21 123 . . 3 𝜑 → (𝜑 → ⊥))
2 ax-1 6 . . . . 5 𝜑 → ((𝜑 → ⊥) → ¬ 𝜑))
3 falim 1554 . . . . 5 (⊥ → ((𝜑 → ⊥) → ¬ 𝜑))
42, 3ja 188 . . . 4 ((𝜑 → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑))
54pm2.43i 52 . . 3 ((𝜑 → ⊥) → ¬ 𝜑)
61, 5impbii 211 . 2 𝜑 ↔ (𝜑 → ⊥))
7 tbw-bijust 1699 . 2 ((¬ 𝜑 ↔ (𝜑 → ⊥)) ↔ (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥))
86, 7mpbi 232 1 (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ⊥wfal 1549 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 209  df-tru 1540  df-fal 1550 This theorem is referenced by:  re1luk2  1712  re1luk3  1713
 Copyright terms: Public domain W3C validator