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Theorem tbw-negdf 1464
 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 pm2.21 100 . . 3 φ → (φ → ⊥ ))
2 ax-1 5 . . . . 5 φ → ((φ → ⊥ ) → ¬ φ))
3 falim 1328 . . . . 5 ( ⊥ → ((φ → ⊥ ) → ¬ φ))
42, 3ja 153 . . . 4 ((φ → ⊥ ) → ((φ → ⊥ ) → ¬ φ))
54pm2.43i 43 . . 3 ((φ → ⊥ ) → ¬ φ)
61, 5impbii 180 . 2 φ ↔ (φ → ⊥ ))
7 tbw-bijust 1463 . 2 ((¬ φ ↔ (φ → ⊥ )) ↔ (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ))
86, 7mpbi 199 1 (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ⊥ wfal 1317 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320 This theorem is referenced by:  re1luk2  1476  re1luk3  1477
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