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Theorem mercolem6 1780
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1774. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem6 ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))

Proof of Theorem mercolem6
StepHypRef Expression
1 merco2 1774 . 2 (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))
2 mercolem1 1775 . . . . . . . 8 (((𝜑 → (𝜑 → (𝜓 → (𝜑𝜒)))) → (𝜑𝜒)) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))
3 mercolem1 1775 . . . . . . . 8 ((((𝜑 → (𝜑 → (𝜓 → (𝜑𝜒)))) → (𝜑𝜒)) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))) → ((𝜑𝜒) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))
42, 3ax-mp 5 . . . . . . 7 ((𝜑𝜒) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))
5 mercolem5 1779 . . . . . . . 8 (𝜑 → ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))
6 mercolem4 1778 . . . . . . . 8 ((𝜑 → ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))) → (((𝜑𝜒) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))))
75, 6ax-mp 5 . . . . . . 7 (((𝜑𝜒) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))))
84, 7ax-mp 5 . . . . . 6 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))
91, 8ax-mp 5 . . . . 5 ((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))
10 mercolem1 1775 . . . . . . . 8 (((𝜑 → (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))
11 mercolem1 1775 . . . . . . . 8 ((((𝜑 → (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))))
1210, 11ax-mp 5 . . . . . . 7 (((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))))
13 mercolem5 1779 . . . . . . . 8 ((𝜑 → (𝜓 → (𝜑𝜒))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))))
14 mercolem4 1778 . . . . . . . 8 (((𝜑 → (𝜓 → (𝜑𝜒))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))))) → ((((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))))))
1513, 14ax-mp 5 . . . . . . 7 ((((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))))))
1612, 15ax-mp 5 . . . . . 6 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))))
171, 16ax-mp 5 . . . . 5 (((𝜑 → (𝜓 → (𝜑𝜒))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))))
189, 17ax-mp 5 . . . 4 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))))
191, 18ax-mp 5 . . 3 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))))
201, 19ax-mp 5 . 2 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒))))
211, 20ax-mp 5 1 ((𝜑 → (𝜓 → (𝜑𝜒))) → (𝜓 → (𝜑𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1601
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 197  df-tru 1599  df-fal 1602
This theorem is referenced by:  mercolem7  1781  re1tbw1  1783  re1tbw2  1784  re1tbw3  1785  pm2.43bgbi  39142
  Copyright terms: Public domain W3C validator