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Theorem tbwlem1 1470
 Description: Used to rederive the Lukasiewicz axioms from Tarski-Bernays-Wajsberg'. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
tbwlem1 ((φ → (ψχ)) → (ψ → (φχ)))

Proof of Theorem tbwlem1
StepHypRef Expression
1 tbw-ax2 1466 . . . 4 (ψ → ((ψχ) → ψ))
2 tbw-ax1 1465 . . . 4 (((ψχ) → ψ) → ((ψχ) → ((ψχ) → χ)))
31, 2tbwsyl 1469 . . 3 (ψ → ((ψχ) → ((ψχ) → χ)))
4 tbw-ax1 1465 . . . 4 (((ψχ) → ((ψχ) → χ)) → ((((ψχ) → χ) → χ) → ((ψχ) → χ)))
5 tbw-ax3 1467 . . . 4 (((((ψχ) → χ) → χ) → ((ψχ) → χ)) → ((ψχ) → χ))
64, 5tbwsyl 1469 . . 3 (((ψχ) → ((ψχ) → χ)) → ((ψχ) → χ))
73, 6tbwsyl 1469 . 2 (ψ → ((ψχ) → χ))
8 tbw-ax1 1465 . 2 ((φ → (ψχ)) → (((ψχ) → χ) → (φχ)))
9 tbw-ax1 1465 . 2 ((ψ → ((ψχ) → χ)) → ((((ψχ) → χ) → (φχ)) → (ψ → (φχ))))
107, 8, 9mpsyl 59 1 ((φ → (ψχ)) → (ψ → (φχ)))
 Colors of variables: wff setvar class Syntax hints:   → 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:  tbwlem2  1471  tbwlem4  1473  tbwlem5  1474  re1luk3  1477
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