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Theorem luklem4 1426
 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 1421 . . . 4 ((¬ ((¬ φφ) → φ) → ((¬ φφ) → φ)) → ((¬ φφ) → φ))
2 luk-2 1421 . . . . 5 ((¬ φφ) → φ)
3 luklem3 1425 . . . . 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 1420 . . 3 ((¬ ψ → ((¬ φφ) → φ)) → ((((¬ φφ) → φ) → ψ) → (¬ ψψ)))
75, 6ax-mp 5 . 2 ((((¬ φφ) → φ) → ψ) → (¬ ψψ))
8 luk-2 1421 . 2 ((¬ ψψ) → ψ)
97, 8luklem1 1423 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-meredith 1406 This theorem is referenced by:  luklem5  1427  luklem6  1428  ax3  1433
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