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Theorem luklem7 1580
 Description: Used to rederive standard propositional axioms from Lukasiewicz'. (Contributed by NM, 22-Dec-2002.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
luklem7 ((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))

Proof of Theorem luklem7
StepHypRef Expression
1 luk-1 1571 . 2 ((𝜑 → (𝜓𝜒)) → (((𝜓𝜒) → 𝜒) → (𝜑𝜒)))
2 luklem5 1578 . . . . 5 (𝜓 → ((𝜓𝜒) → 𝜓))
3 luk-1 1571 . . . . 5 (((𝜓𝜒) → 𝜓) → ((𝜓𝜒) → ((𝜓𝜒) → 𝜒)))
42, 3luklem1 1574 . . . 4 (𝜓 → ((𝜓𝜒) → ((𝜓𝜒) → 𝜒)))
5 luklem6 1579 . . . 4 (((𝜓𝜒) → ((𝜓𝜒) → 𝜒)) → ((𝜓𝜒) → 𝜒))
64, 5luklem1 1574 . . 3 (𝜓 → ((𝜓𝜒) → 𝜒))
7 luk-1 1571 . . 3 ((𝜓 → ((𝜓𝜒) → 𝜒)) → ((((𝜓𝜒) → 𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒))))
86, 7ax-mp 5 . 2 ((((𝜓𝜒) → 𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒)))
91, 8luklem1 1574 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:  luklem8  1581  ax2  1583
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