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Theorem mercolem7 1781
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
mercolem7 ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))

Proof of Theorem mercolem7
StepHypRef Expression
1 merco2 1774 . 2 (((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑))))
2 mercolem3 1777 . . . 4 (((𝜑𝜒) → (𝜃𝜓)) → ((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))))
3 mercolem6 1780 . . . 4 ((((𝜑𝜒) → (𝜃𝜓)) → ((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))) → ((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))))
42, 3ax-mp 5 . . 3 ((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))
5 mercolem5 1779 . . . 4 (𝜑 → ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))))
6 mercolem4 1778 . . . 4 ((𝜑 → ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))) → (((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))))))
75, 6ax-mp 5 . . 3 (((𝜑𝜒) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))) → ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))))
84, 7ax-mp 5 . 2 ((((𝜑𝜑) → ((⊥ → 𝜑) → 𝜑)) → ((𝜑𝜑) → (𝜑 → (𝜑𝜑)))) → ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓))))
91, 8ax-mp 5 1 ((𝜑𝜓) → (((𝜑𝜒) → (𝜃𝜓)) → (𝜃𝜓)))
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:  mercolem8  1782
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