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Theorem retbwax1 1500
 Description: tbw-ax1 1465 rederived from merco1 1478. This theorem, along with retbwax2 1481, retbwax3 1488, and retbwax4 1480, shows that merco1 1478 with ax-mp 5 can be used as a complete axiomatization of propositional calculus. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
retbwax1 ((φψ) → ((ψχ) → (φχ)))

Proof of Theorem retbwax1
StepHypRef Expression
1 merco1lem18 1499 . . 3 ((ψ → (φχ)) → ((φψ) → (φχ)))
2 merco1lem16 1497 . . 3 (((ψ → (φχ)) → ((φψ) → (φχ))) → ((ψχ) → ((φψ) → (φχ))))
31, 2ax-mp 5 . 2 ((ψχ) → ((φψ) → (φχ)))
4 merco1lem15 1496 . . . . . 6 (((φψ) → (φχ)) → ((φψ) → ((ψχ) → (φχ))))
5 merco1lem15 1496 . . . . . 6 ((((φψ) → (φχ)) → ((φψ) → ((ψχ) → (φχ)))) → (((φψ) → (φχ)) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))
64, 5ax-mp 5 . . . . 5 (((φψ) → (φχ)) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))
7 merco1lem18 1499 . . . . 5 ((((φψ) → (φχ)) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ((((ψχ) → ((φψ) → (φχ))) → ((φψ) → (φχ))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))
86, 7ax-mp 5 . . . 4 ((((ψχ) → ((φψ) → (φχ))) → ((φψ) → (φχ))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))
9 merco1lem14 1495 . . . 4 (((((ψχ) → ((φψ) → (φχ))) → ((φψ) → (φχ))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))
108, 9ax-mp 5 . . 3 ((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))
11 merco1lem14 1495 . . . . . 6 (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ)) → ((ψχ) → (φχ)))
12 merco1lem10 1491 . . . . . . . . 9 (((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ ) → ⊥ ) → ((((ψχ) → (φχ)) → φ) → ⊥ )) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )) → (((φχ) → φ) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )))
13 merco1 1478 . . . . . . . . 9 ((((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ ) → ⊥ ) → ((((ψχ) → (φχ)) → φ) → ⊥ )) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )) → (((φχ) → φ) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ))) → (((((φχ) → φ) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → ((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ ))))
1412, 13ax-mp 5 . . . . . . . 8 (((((φχ) → φ) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → ((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )))
15 merco1 1478 . . . . . . . 8 ((((((φχ) → φ) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ )) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → ((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ ))) → ((((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → (φχ)) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ))))
1614, 15ax-mp 5 . . . . . . 7 ((((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → (φχ)) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ)))
17 merco1 1478 . . . . . . 7 (((((((ψχ) → (φχ)) → φ) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ⊥ )) → (φχ)) → ((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ))) → ((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ)) → ((ψχ) → (φχ))) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((ψχ) → (φχ)))))
1816, 17ax-mp 5 . . . . . 6 ((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (φχ)) → ((ψχ) → (φχ))) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((ψχ) → (φχ))))
1911, 18ax-mp 5 . . . . 5 (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((ψχ) → (φχ)))
20 merco1lem15 1496 . . . . 5 ((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((ψχ) → (φχ))) → (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))))
2119, 20ax-mp 5 . . . 4 (((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ))))
22 merco1lem10 1491 . . . . . 6 ((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → ((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))))
23 merco1lem9 1490 . . . . . 6 (((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → ((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))) → ((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))))
2422, 23ax-mp 5 . . . . 5 ((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))
25 merco1lem13 1494 . . . . 5 (((((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))) → ((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))))
2624, 25ax-mp 5 . . . 4 ((((((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → ⊥ ) → ((φψ) → ((ψχ) → (φχ)))) → (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))))
2721, 26ax-mp 5 . . 3 (((ψχ) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))) → (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ)))))
2810, 27ax-mp 5 . 2 (((ψχ) → ((φψ) → (φχ))) → ((φψ) → ((ψχ) → (φχ))))
293, 28ax-mp 5 1 ((φψ) → ((ψχ) → (φχ)))
 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: (None)
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