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Theorem re1luk3 1677
Description: luk-3 1622 derived from the Tarski-Bernays-Wajsberg axioms.

This theorem, along with re1luk1 1675 and re1luk2 1676 proves that tbw-ax1 1665, tbw-ax2 1666, tbw-ax3 1667, and tbw-ax4 1668, with ax-mp 5 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
re1luk3 (𝜑 → (¬ 𝜑𝜓))

Proof of Theorem re1luk3
StepHypRef Expression
1 tbw-negdf 1664 . . 3 (((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥)
2 tbwlem5 1674 . . 3 ((((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → ¬ 𝜑) → ⊥)) → ⊥) → (¬ 𝜑 → (𝜑 → ⊥)))
31, 2ax-mp 5 . 2 𝜑 → (𝜑 → ⊥))
4 tbw-ax4 1668 . . . 4 (⊥ → 𝜓)
5 tbw-ax1 1665 . . . . 5 ((𝜑 → ⊥) → ((⊥ → 𝜓) → (𝜑𝜓)))
6 tbwlem1 1670 . . . . 5 (((𝜑 → ⊥) → ((⊥ → 𝜓) → (𝜑𝜓))) → ((⊥ → 𝜓) → ((𝜑 → ⊥) → (𝜑𝜓))))
75, 6ax-mp 5 . . . 4 ((⊥ → 𝜓) → ((𝜑 → ⊥) → (𝜑𝜓)))
84, 7ax-mp 5 . . 3 ((𝜑 → ⊥) → (𝜑𝜓))
9 tbwlem1 1670 . . 3 (((𝜑 → ⊥) → (𝜑𝜓)) → (𝜑 → ((𝜑 → ⊥) → 𝜓)))
108, 9ax-mp 5 . 2 (𝜑 → ((𝜑 → ⊥) → 𝜓))
11 tbw-ax1 1665 . 2 ((¬ 𝜑 → (𝜑 → ⊥)) → (((𝜑 → ⊥) → 𝜓) → (¬ 𝜑𝜓)))
123, 10, 11mpsyl 68 1 (𝜑 → (¬ 𝜑𝜓))
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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