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Theorem mercolem2 1503
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1501. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem2 (((φψ) → φ) → (χ → (θφ)))

Proof of Theorem mercolem2
StepHypRef Expression
1 merco2 1501 . 2 (((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ))))
2 merco2 1501 . . . 4 (((φφ) → (( ⊥ → φ) → (φψ))) → (((φψ) → φ) → (χ → (θφ))))
3 merco2 1501 . . . . . . . 8 (((φψ) → (( ⊥ → φ) → ⊥ )) → (( ⊥ → φ) → (χ → (θφ))))
4 merco2 1501 . . . . . . . 8 ((((φψ) → (( ⊥ → φ) → ⊥ )) → (( ⊥ → φ) → (χ → (θφ)))) → (((χ → (θφ)) → (φψ)) → (( ⊥ → φ) → (( ⊥ → φ) → (φψ)))))
53, 4ax-mp 5 . . . . . . 7 (((χ → (θφ)) → (φψ)) → (( ⊥ → φ) → (( ⊥ → φ) → (φψ))))
6 merco2 1501 . . . . . . 7 ((((χ → (θφ)) → (φψ)) → (( ⊥ → φ) → (( ⊥ → φ) → (φψ)))) → (((( ⊥ → φ) → (φψ)) → (χ → (θφ))) → (( ⊥ → φ) → (((φψ) → φ) → (χ → (θφ))))))
75, 6ax-mp 5 . . . . . 6 (((( ⊥ → φ) → (φψ)) → (χ → (θφ))) → (( ⊥ → φ) → (((φψ) → φ) → (χ → (θφ)))))
8 merco2 1501 . . . . . 6 ((((( ⊥ → φ) → (φψ)) → (χ → (θφ))) → (( ⊥ → φ) → (((φψ) → φ) → (χ → (θφ))))) → (((((φψ) → φ) → (χ → (θφ))) → (( ⊥ → φ) → (φψ))) → (( ⊥ → φ) → ((φφ) → (( ⊥ → φ) → (φψ))))))
97, 8ax-mp 5 . . . . 5 (((((φψ) → φ) → (χ → (θφ))) → (( ⊥ → φ) → (φψ))) → (( ⊥ → φ) → ((φφ) → (( ⊥ → φ) → (φψ)))))
10 merco2 1501 . . . . 5 ((((((φψ) → φ) → (χ → (θφ))) → (( ⊥ → φ) → (φψ))) → (( ⊥ → φ) → ((φφ) → (( ⊥ → φ) → (φψ))))) → ((((φφ) → (( ⊥ → φ) → (φψ))) → (((φψ) → φ) → (χ → (θφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φψ) → φ) → (χ → (θφ)))))))
119, 10ax-mp 5 . . . 4 ((((φφ) → (( ⊥ → φ) → (φψ))) → (((φψ) → φ) → (χ → (θφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φψ) → φ) → (χ → (θφ))))))
122, 11ax-mp 5 . . 3 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φψ) → φ) → (χ → (θφ)))))
131, 12ax-mp 5 . 2 ((((φφ) → (( ⊥ → φ) → φ)) → ((φφ) → (φ → (φφ)))) → (((φψ) → φ) → (χ → (θφ))))
141, 13ax-mp 5 1 (((φψ) → φ) → (χ → (θφ)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 177  df-tru 1319  df-fal 1320
This theorem is referenced by:  mercolem3  1504  mercolem5  1506  re1tbw3  1512
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