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Theorem luklem6 1428
 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 1420 . 2 ((φ → (φψ)) → (((φψ) → ψ) → (φψ)))
2 luklem5 1427 . . . . . 6 (¬ (φψ) → (¬ ψ → ¬ (φψ)))
3 luklem2 1424 . . . . . . 7 ((¬ ψ → ¬ (φψ)) → (((¬ ψψ) → ψ) → ((φψ) → ψ)))
4 luklem4 1426 . . . . . . 7 ((((¬ ψψ) → ψ) → ((φψ) → ψ)) → ((φψ) → ψ))
53, 4luklem1 1423 . . . . . 6 ((¬ ψ → ¬ (φψ)) → ((φψ) → ψ))
62, 5luklem1 1423 . . . . 5 (¬ (φψ) → ((φψ) → ψ))
7 luk-1 1420 . . . . 5 ((¬ (φψ) → ((φψ) → ψ)) → ((((φψ) → ψ) → (φψ)) → (¬ (φψ) → (φψ))))
86, 7ax-mp 5 . . . 4 ((((φψ) → ψ) → (φψ)) → (¬ (φψ) → (φψ)))
9 luk-1 1420 . . . 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 1426 . . 3 ((((¬ (φψ) → (φψ)) → (φψ)) → ((((φψ) → ψ) → (φψ)) → (φψ))) → ((((φψ) → ψ) → (φψ)) → (φψ)))
1210, 11ax-mp 5 . 2 ((((φψ) → ψ) → (φψ)) → (φψ))
131, 12luklem1 1423 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-meredith 1406 This theorem is referenced by:  luklem7  1429  ax2  1432
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