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Theorem tbwlem1 1670
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
tbwlem1 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Proof of Theorem tbwlem1
StepHypRef Expression
1 tbw-ax2 1666 . . . 4 (𝜓 → ((𝜓𝜒) → 𝜓))
2 tbw-ax1 1665 . . . 4 (((𝜓𝜒) → 𝜓) → ((𝜓𝜒) → ((𝜓𝜒) → 𝜒)))
31, 2tbwsyl 1669 . . 3 (𝜓 → ((𝜓𝜒) → ((𝜓𝜒) → 𝜒)))
4 tbw-ax1 1665 . . . 4 (((𝜓𝜒) → ((𝜓𝜒) → 𝜒)) → ((((𝜓𝜒) → 𝜒) → 𝜒) → ((𝜓𝜒) → 𝜒)))
5 tbw-ax3 1667 . . . 4 (((((𝜓𝜒) → 𝜒) → 𝜒) → ((𝜓𝜒) → 𝜒)) → ((𝜓𝜒) → 𝜒))
64, 5tbwsyl 1669 . . 3 (((𝜓𝜒) → ((𝜓𝜒) → 𝜒)) → ((𝜓𝜒) → 𝜒))
73, 6tbwsyl 1669 . 2 (𝜓 → ((𝜓𝜒) → 𝜒))
8 tbw-ax1 1665 . 2 ((𝜑 → (𝜓𝜒)) → (((𝜓𝜒) → 𝜒) → (𝜑𝜒)))
9 tbw-ax1 1665 . 2 ((𝜓 → ((𝜓𝜒) → 𝜒)) → ((((𝜓𝜒) → 𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒))))
107, 8, 9mpsyl 68 1 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem is referenced by:  tbwlem2  1671  tbwlem4  1673  tbwlem5  1674  re1luk3  1677
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