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Theorem luklem2 1624
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
luklem2 ((𝜑 → ¬ 𝜓) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))

Proof of Theorem luklem2
StepHypRef Expression
1 luk-1 1620 . . 3 ((𝜑 → ¬ 𝜓) → ((¬ 𝜓𝜒) → (𝜑𝜒)))
2 luk-3 1622 . . . 4 (𝜓 → (¬ 𝜓𝜒))
3 luk-1 1620 . . . 4 ((𝜓 → (¬ 𝜓𝜒)) → (((¬ 𝜓𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒))))
42, 3ax-mp 5 . . 3 (((¬ 𝜓𝜒) → (𝜑𝜒)) → (𝜓 → (𝜑𝜒)))
51, 4luklem1 1623 . 2 ((𝜑 → ¬ 𝜓) → (𝜓 → (𝜑𝜒)))
6 luk-1 1620 . 2 ((𝜓 → (𝜑𝜒)) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
75, 6luklem1 1623 1 ((𝜑 → ¬ 𝜓) → (((𝜑𝜒) → 𝜃) → (𝜓𝜃)))
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:  luklem3  1625  luklem6  1628  ax3  1633
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