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Theorem mercolem4 1742
 Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1738. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem4 ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))

Proof of Theorem mercolem4
StepHypRef Expression
1 merco2 1738 . 2 (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))
2 merco2 1738 . . . 4 ((((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))
3 merco2 1738 . . . . . . . . 9 (((𝜑𝜑) → ((⊥ → 𝜑) → (𝜃𝜒))) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))
4 mercolem1 1739 . . . . . . . . 9 ((((𝜑𝜑) → ((⊥ → 𝜑) → (𝜃𝜒))) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → (((⊥ → 𝜑) → (𝜃𝜒)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))))
53, 4ax-mp 5 . . . . . . . 8 (((⊥ → 𝜑) → (𝜃𝜒)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))
6 mercolem1 1739 . . . . . . . 8 ((((⊥ → 𝜑) → (𝜃𝜒)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))) → ((𝜃𝜒) → ((⊥ → 𝜑) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))))
75, 6ax-mp 5 . . . . . . 7 ((𝜃𝜒) → ((⊥ → 𝜑) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))))
8 merco2 1738 . . . . . . 7 (((𝜃𝜒) → ((⊥ → 𝜑) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))) → ((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃))))
97, 8ax-mp 5 . . . . . 6 ((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)))
10 mercolem3 1741 . . . . . 6 (((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃))) → ((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → ((⊥ → 𝜑) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)))))
119, 10ax-mp 5 . . . . 5 ((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → ((⊥ → 𝜑) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃))))
12 merco2 1738 . . . . 5 (((((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))) → 𝜃) → ((⊥ → 𝜑) → (((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)))) → (((((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))))))
1311, 12ax-mp 5 . . . 4 (((((𝜂𝜑) → 𝜑) → ((⊥ → 𝜑) → 𝜃)) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))))
142, 13ax-mp 5 . . 3 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))))
151, 14ax-mp 5 . 2 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑)))))
161, 15ax-mp 5 1 ((𝜃 → (𝜂𝜑)) → (((𝜃𝜒) → 𝜑) → (𝜏 → (𝜂𝜑))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ⊥wfal 1550 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 210  df-tru 1541  df-fal 1551 This theorem is referenced by:  mercolem6  1744  mercolem7  1745
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