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Theorem re1luk3 1477
 Description: luk-3 1422 derived from the Tarski-Bernays-Wajsberg axioms. This theorem, along with re1luk1 1475 and re1luk2 1476 proves that tbw-ax1 1465, tbw-ax2 1466, tbw-ax3 1467, and tbw-ax4 1468, with ax-mp 8 can be used as a complete axiom system for all of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1luk3 (φ → (¬ φψ))

Proof of Theorem re1luk3
StepHypRef Expression
1 tbw-negdf 1464 . . 3 (((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ )
2 tbwlem5 1474 . . 3 ((((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ¬ φ) → ⊥ )) → ⊥ ) → (¬ φ → (φ → ⊥ )))
31, 2ax-mp 8 . 2 φ → (φ → ⊥ ))
4 tbw-ax4 1468 . . . 4 ( ⊥ → ψ)
5 tbw-ax1 1465 . . . . 5 ((φ → ⊥ ) → (( ⊥ → ψ) → (φψ)))
6 tbwlem1 1470 . . . . 5 (((φ → ⊥ ) → (( ⊥ → ψ) → (φψ))) → (( ⊥ → ψ) → ((φ → ⊥ ) → (φψ))))
75, 6ax-mp 8 . . . 4 (( ⊥ → ψ) → ((φ → ⊥ ) → (φψ)))
84, 7ax-mp 8 . . 3 ((φ → ⊥ ) → (φψ))
9 tbwlem1 1470 . . 3 (((φ → ⊥ ) → (φψ)) → (φ → ((φ → ⊥ ) → ψ)))
108, 9ax-mp 8 . 2 (φ → ((φ → ⊥ ) → ψ))
11 tbw-ax1 1465 . 2 ((¬ φ → (φ → ⊥ )) → (((φ → ⊥ ) → ψ) → (¬ φψ)))
123, 10, 11mpsyl 59 1 (φ → (¬ φψ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ⊥ wfal 1317 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320 This theorem is referenced by: (None)
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