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Theorem re1luk2 1676
Description: luk-2 1621 derived from the Tarski-Bernays-Wajsberg axioms. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1luk2 ((¬ 𝜑𝜑) → 𝜑)

Proof of Theorem re1luk2
StepHypRef Expression
1 tbw-negdf 1664 . . . 4 (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
2 tbw-ax2 1666 . . . . 5 ((((𝜑 → ⊥) → ¬ 𝜑) → ⊥) → ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)))
3 tbwlem4 1673 . . . . 5 (((((𝜑 → ⊥) → ¬ 𝜑) → ⊥) → ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥))) → ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑)))
42, 3ax-mp 5 . . . 4 ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → ((𝜑 → ⊥) → ¬ 𝜑))
51, 4ax-mp 5 . . 3 ((𝜑 → ⊥) → ¬ 𝜑)
6 tbw-ax1 1665 . . 3 (((𝜑 → ⊥) → ¬ 𝜑) → ((¬ 𝜑𝜑) → ((𝜑 → ⊥) → 𝜑)))
75, 6ax-mp 5 . 2 ((¬ 𝜑𝜑) → ((𝜑 → ⊥) → 𝜑))
8 tbw-ax3 1667 . 2 (((𝜑 → ⊥) → 𝜑) → 𝜑)
97, 8tbwsyl 1669 1 ((¬ 𝜑𝜑) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1528
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 1526  df-fal 1529
This theorem is referenced by: (None)
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