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Theorem mercolem1 1775
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
mercolem1 (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))

Proof of Theorem mercolem1
StepHypRef Expression
1 merco2 1774 . 2 (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))
2 merco2 1774 . . . 4 (((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))))
3 merco2 1774 . . . . . . 7 (((𝜓 → (𝜃𝜒)) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜓) → ((⊥ → 𝜑) → (𝜑𝜓))))
4 merco2 1774 . . . . . . 7 ((((𝜓 → (𝜃𝜒)) → ((⊥ → 𝜑) → ⊥)) → ((⊥ → 𝜓) → ((⊥ → 𝜑) → (𝜑𝜓)))) → ((((⊥ → 𝜑) → (𝜑𝜓)) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))))))
53, 4ax-mp 5 . . . . . 6 ((((⊥ → 𝜑) → (𝜑𝜓)) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))))
6 merco2 1774 . . . . . 6 (((((⊥ → 𝜑) → (𝜑𝜓)) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))))) → (((((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (𝜑𝜓))) → ((⊥ → 𝜑) → ((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓))))))
75, 6ax-mp 5 . . . . 5 (((((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (𝜑𝜓))) → ((⊥ → 𝜑) → ((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓)))))
8 merco2 1774 . . . . 5 ((((((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))) → ((⊥ → 𝜑) → (𝜑𝜓))) → ((⊥ → 𝜑) → ((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓))))) → ((((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))))))
97, 8ax-mp 5 . . . 4 ((((𝜒𝜑) → ((⊥ → 𝜑) → (𝜑𝜓))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))))))
102, 9ax-mp 5 . . 3 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))))
111, 10ax-mp 5 . 2 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒))))
121, 11ax-mp 5 1 (((𝜑𝜓) → 𝜒) → (𝜓 → (𝜃𝜒)))
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:  mercolem4  1778  mercolem5  1779  mercolem6  1780  re1tbw2  1784
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