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Theorem impsingle 1630
 Description: The shortest single axiom for implicational calculus, due to Lukasiewicz. It is now known to be the unique shortest axiom. The axiom is proved here starting from { ax-1 6, ax-2 7 and peirce 205 }. (Contributed by Larry Lesyna and Jeffrey P. Machado, 18-Jul-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
impsingle (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → (𝜃𝜑)))

Proof of Theorem impsingle
StepHypRef Expression
1 imim1 83 . 2 (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → ((𝜑𝜓) → 𝜑)))
2 peirce 205 . . 3 (((𝜑𝜓) → 𝜑) → 𝜑)
32a1d 25 . 2 (((𝜑𝜓) → 𝜑) → (𝜃𝜑))
41, 3syl6 35 1 (((𝜑𝜓) → 𝜒) → ((𝜒𝜑) → (𝜃𝜑)))
 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:  impsingle-step4  1631  impsingle-step8  1632  impsingle-step15  1634  impsingle-step18  1635  impsingle-step20  1637  impsingle-step22  1639
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