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Theorem re1tbw4 1513
 Description: tbw-ax4 1468 rederived from merco2 1501. This theorem, along with re1tbw1 1510, re1tbw2 1511, and re1tbw3 1512, shows that merco2 1501, along with ax-mp 8, can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
re1tbw4 ( ⊥ → φ)

Proof of Theorem re1tbw4
StepHypRef Expression
1 re1tbw3 1512 . . 3 (((φφ) → φ) → φ)
2 re1tbw2 1511 . . . 4 (φ → ((φφ) → φ))
3 re1tbw1 1510 . . . 4 ((φ → ((φφ) → φ)) → ((((φφ) → φ) → φ) → (φφ)))
42, 3ax-mp 8 . . 3 ((((φφ) → φ) → φ) → (φφ))
51, 4ax-mp 8 . 2 (φφ)
6 re1tbw3 1512 . . . . 5 (((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ))
7 re1tbw2 1511 . . . . . 6 (( ⊥ → φ) → ((( ⊥ → φ) → φ) → ( ⊥ → φ)))
8 re1tbw1 1510 . . . . . 6 ((( ⊥ → φ) → ((( ⊥ → φ) → φ) → ( ⊥ → φ))) → ((((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ)) → (( ⊥ → φ) → ( ⊥ → φ))))
97, 8ax-mp 8 . . . . 5 ((((( ⊥ → φ) → φ) → ( ⊥ → φ)) → ( ⊥ → φ)) → (( ⊥ → φ) → ( ⊥ → φ)))
106, 9ax-mp 8 . . . 4 (( ⊥ → φ) → ( ⊥ → φ))
11 mercolem3 1504 . . . . 5 ((( ⊥ → φ) → φ) → (( ⊥ → φ) → ( ⊥ → φ)))
12 merco2 1501 . . . . 5 (((( ⊥ → φ) → φ) → (( ⊥ → φ) → ( ⊥ → φ))) → ((( ⊥ → φ) → ( ⊥ → φ)) → ((φφ) → ((φφ) → ( ⊥ → φ)))))
1311, 12ax-mp 8 . . . 4 ((( ⊥ → φ) → ( ⊥ → φ)) → ((φφ) → ((φφ) → ( ⊥ → φ))))
1410, 13ax-mp 8 . . 3 ((φφ) → ((φφ) → ( ⊥ → φ)))
155, 14ax-mp 8 . 2 ((φφ) → ( ⊥ → φ))
165, 15ax-mp 8 1 ( ⊥ → φ)
 Colors of variables: wff setvar class Syntax hints:   → 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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