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Theorem luklem4 1577
 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
luklem4 ((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → 𝜓)

Proof of Theorem luklem4
StepHypRef Expression
1 luk-2 1572 . . . 4 ((¬ ((¬ 𝜑𝜑) → 𝜑) → ((¬ 𝜑𝜑) → 𝜑)) → ((¬ 𝜑𝜑) → 𝜑))
2 luk-2 1572 . . . . 5 ((¬ 𝜑𝜑) → 𝜑)
3 luklem3 1576 . . . . 5 (((¬ 𝜑𝜑) → 𝜑) → (((¬ ((¬ 𝜑𝜑) → 𝜑) → ((¬ 𝜑𝜑) → 𝜑)) → ((¬ 𝜑𝜑) → 𝜑)) → (¬ 𝜓 → ((¬ 𝜑𝜑) → 𝜑))))
42, 3ax-mp 5 . . . 4 (((¬ ((¬ 𝜑𝜑) → 𝜑) → ((¬ 𝜑𝜑) → 𝜑)) → ((¬ 𝜑𝜑) → 𝜑)) → (¬ 𝜓 → ((¬ 𝜑𝜑) → 𝜑)))
51, 4ax-mp 5 . . 3 𝜓 → ((¬ 𝜑𝜑) → 𝜑))
6 luk-1 1571 . . 3 ((¬ 𝜓 → ((¬ 𝜑𝜑) → 𝜑)) → ((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → (¬ 𝜓𝜓)))
75, 6ax-mp 5 . 2 ((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → (¬ 𝜓𝜓))
8 luk-2 1572 . 2 ((¬ 𝜓𝜓) → 𝜓)
97, 8luklem1 1574 1 ((((¬ 𝜑𝜑) → 𝜑) → 𝜓) → 𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → 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:  luklem5  1578  luklem6  1579  ax3  1584
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