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Theorem luklem6 1579
 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
luklem6 ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem luklem6
StepHypRef Expression
1 luk-1 1571 . 2 ((𝜑 → (𝜑𝜓)) → (((𝜑𝜓) → 𝜓) → (𝜑𝜓)))
2 luklem5 1578 . . . . . 6 (¬ (𝜑𝜓) → (¬ 𝜓 → ¬ (𝜑𝜓)))
3 luklem2 1575 . . . . . . 7 ((¬ 𝜓 → ¬ (𝜑𝜓)) → (((¬ 𝜓𝜓) → 𝜓) → ((𝜑𝜓) → 𝜓)))
4 luklem4 1577 . . . . . . 7 ((((¬ 𝜓𝜓) → 𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓))
53, 4luklem1 1574 . . . . . 6 ((¬ 𝜓 → ¬ (𝜑𝜓)) → ((𝜑𝜓) → 𝜓))
62, 5luklem1 1574 . . . . 5 (¬ (𝜑𝜓) → ((𝜑𝜓) → 𝜓))
7 luk-1 1571 . . . . 5 ((¬ (𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (¬ (𝜑𝜓) → (𝜑𝜓))))
86, 7ax-mp 5 . . . 4 ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (¬ (𝜑𝜓) → (𝜑𝜓)))
9 luk-1 1571 . . . 4 (((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (¬ (𝜑𝜓) → (𝜑𝜓))) → (((¬ (𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓)) → ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (𝜑𝜓))))
108, 9ax-mp 5 . . 3 (((¬ (𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓)) → ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (𝜑𝜓)))
11 luklem4 1577 . . 3 ((((¬ (𝜑𝜓) → (𝜑𝜓)) → (𝜑𝜓)) → ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (𝜑𝜓))) → ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (𝜑𝜓)))
1210, 11ax-mp 5 . 2 ((((𝜑𝜓) → 𝜓) → (𝜑𝜓)) → (𝜑𝜓))
131, 12luklem1 1574 1 ((𝜑 → (𝜑𝜓)) → (𝜑𝜓))
 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:  luklem7  1580  ax2  1583
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