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Theorem tbwlem2 1671
 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
tbwlem2 ((𝜑 → (𝜓 → ⊥)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))

Proof of Theorem tbwlem2
StepHypRef Expression
1 tbw-ax4 1668 . . . . 5 (⊥ → 𝜒)
2 tbw-ax1 1665 . . . . . 6 ((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓𝜒)))
3 tbwlem1 1670 . . . . . 6 (((𝜓 → ⊥) → ((⊥ → 𝜒) → (𝜓𝜒))) → ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓𝜒))))
42, 3ax-mp 5 . . . . 5 ((⊥ → 𝜒) → ((𝜓 → ⊥) → (𝜓𝜒)))
51, 4ax-mp 5 . . . 4 ((𝜓 → ⊥) → (𝜓𝜒))
6 tbwlem1 1670 . . . 4 (((𝜓 → ⊥) → (𝜓𝜒)) → (𝜓 → ((𝜓 → ⊥) → 𝜒)))
75, 6ax-mp 5 . . 3 (𝜓 → ((𝜓 → ⊥) → 𝜒))
8 tbw-ax1 1665 . . 3 ((𝜑 → (𝜓 → ⊥)) → (((𝜓 → ⊥) → 𝜒) → (𝜑𝜒)))
9 tbw-ax1 1665 . . 3 ((𝜓 → ((𝜓 → ⊥) → 𝜒)) → ((((𝜓 → ⊥) → 𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒))))
107, 8, 9mpsyl 68 . 2 ((𝜑 → (𝜓 → ⊥)) → (𝜓 → (𝜑𝜒)))
11 tbw-ax1 1665 . 2 ((𝜓 → (𝜑𝜒)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
1210, 11tbwsyl 1669 1 ((𝜑 → (𝜓 → ⊥)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
 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:  tbwlem4  1673
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