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Theorem merco1lem18 1499
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1478. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem18 ((φ → (ψχ)) → ((ψφ) → (ψχ)))

Proof of Theorem merco1lem18
StepHypRef Expression
1 merco1 1478 . . . 4 ((((((ψχ) → ψ) → ((ψφ) → ⊥ )) → ((ψχ) → ψ)) → φ) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))
2 merco1lem17 1498 . . . 4 (((((((ψχ) → ψ) → ((ψφ) → ⊥ )) → ((ψχ) → ψ)) → φ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((((ψχ) → ψ) → φ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))
31, 2ax-mp 8 . . 3 ((((ψχ) → ψ) → φ) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))
4 merco1lem17 1498 . . 3 (((((ψχ) → ψ) → φ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))
53, 4ax-mp 8 . 2 ((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))
6 merco1lem5 1485 . . . . . . 7 ((((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ))
7 merco1lem3 1483 . . . . . . 7 (((((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ ) → ⊥ ) → (((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ )) → (((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )))
86, 7ax-mp 8 . . . . . 6 (((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ ))
9 merco1lem5 1485 . . . . . 6 ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )) → (((φ → (ψχ)) → ((ψφ) → (ψχ))) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )))
108, 9ax-mp 8 . . . . 5 (((φ → (ψχ)) → ((ψφ) → (ψχ))) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ ))
11 merco1lem4 1484 . . . . 5 ((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )) → (((ψφ) → (ψχ)) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )))
1210, 11ax-mp 8 . . . 4 (((ψφ) → (ψχ)) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ ))
13 merco1 1478 . . . . 5 (((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → (ψφ)) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))))
14 merco1lem2 1482 . . . . 5 ((((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → (ψφ)) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))) → ((((ψφ) → (ψχ)) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))))
1513, 14ax-mp 8 . . . 4 ((((ψφ) → (ψχ)) → ((((((φ → (ψχ)) → ((ψφ) → (ψχ))) → ⊥ ) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ⊥ )) → ⊥ ) → ⊥ )) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))))
1612, 15ax-mp 8 . . 3 (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))
17 merco1lem9 1490 . . 3 ((((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))) → (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))))
1816, 17ax-mp 8 . 2 (((ψφ) → ((φ → (ψχ)) → ((ψφ) → (ψχ)))) → ((φ → (ψχ)) → ((ψφ) → (ψχ))))
195, 18ax-mp 8 1 ((φ → (ψχ)) → ((ψφ) → (ψχ)))
 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:  retbwax1  1500
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