MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tbwlem5 Structured version   Visualization version   GIF version

Theorem tbwlem5 1674
Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbwlem5 (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)

Proof of Theorem tbwlem5
StepHypRef Expression
1 tbw-ax2 1666 . . . 4 (𝜑 → (𝜓𝜑))
2 tbw-ax1 1665 . . . 4 ((𝜓𝜑) → ((𝜑 → ⊥) → (𝜓 → ⊥)))
31, 2tbwsyl 1669 . . 3 (𝜑 → ((𝜑 → ⊥) → (𝜓 → ⊥)))
4 tbwlem1 1670 . . 3 ((𝜑 → ((𝜑 → ⊥) → (𝜓 → ⊥))) → ((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥))))
53, 4ax-mp 5 . 2 ((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥)))
6 tbwlem4 1673 . 2 (((𝜑 → ⊥) → (𝜑 → (𝜓 → ⊥))) → (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑))
75, 6ax-mp 5 1 (((𝜑 → (𝜓 → ⊥)) → ⊥) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  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:  re1luk3  1677
  Copyright terms: Public domain W3C validator