MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mercolem8 Structured version   Visualization version   GIF version

Theorem mercolem8 1818
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco2 1810. (Contributed by Anthony Hart, 16-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
mercolem8 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))

Proof of Theorem mercolem8
StepHypRef Expression
1 merco2 1810 . 2 (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))
2 merco2 1810 . . . . 5 ((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))
3 mercolem3 1813 . . . . 5 (((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))) → ((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))))
42, 3ax-mp 5 . . . 4 ((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))))
5 mercolem7 1817 . . . . . 6 ((𝜑𝜓) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))
6 mercolem7 1817 . . . . . 6 (((𝜑𝜓) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓))) → ((((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))))
75, 6ax-mp 5 . . . . 5 ((((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓))))
8 merco2 1810 . . . . 5 (((((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → ((⊥ → 𝜑) → (((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)))) → (((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))))))
97, 8ax-mp 5 . . . 4 (((((𝜑𝜒) → ((⊥ → 𝜑) → 𝜓)) → ((⊥ → 𝜑) → 𝜓)) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))))))
104, 9ax-mp 5 . . 3 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))))
111, 10ax-mp 5 . 2 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒))))))
121, 11ax-mp 5 1 ((𝜑𝜓) → ((𝜓 → (𝜑𝜒)) → (𝜏 → (𝜃 → (𝜑𝜒)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1637
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 1635  df-fal 1638
This theorem is referenced by:  re1tbw1  1819
  Copyright terms: Public domain W3C validator