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Theorem merco1lem14 1695
Description: Used to rederive the Tarski-Bernays-Wajsberg axioms from merco1 1678. (Contributed by Anthony Hart, 18-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
merco1lem14 ((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒))

Proof of Theorem merco1lem14
StepHypRef Expression
1 merco1lem13 1694 . . . 4 ((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓)))
2 merco1lem8 1689 . . . . . 6 (((((𝜑 → ((𝜑𝜓) → 𝜓)) → 𝜑) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → ⊥)) → 𝜑) → (((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)))
3 merco1 1678 . . . . . 6 ((((((𝜑 → ((𝜑𝜓) → 𝜓)) → 𝜑) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → ⊥)) → 𝜑) → (((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓))) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (𝜑 → ((𝜑𝜓) → 𝜓)))))
42, 3ax-mp 5 . . . . 5 (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (𝜑 → ((𝜑𝜓) → 𝜓))))
5 merco1lem9 1690 . . . . 5 ((((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (𝜑 → ((𝜑𝜓) → 𝜓)))) → (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (𝜑 → ((𝜑𝜓) → 𝜓))))
64, 5ax-mp 5 . . . 4 (((((𝜑𝜓) → ((𝜑𝜓) → 𝜓)) → ((𝜑𝜓) → 𝜓)) → (𝜑 → ((𝜑𝜓) → 𝜓))) → (𝜑 → ((𝜑𝜓) → 𝜓)))
71, 6ax-mp 5 . . 3 (𝜑 → ((𝜑𝜓) → 𝜓))
8 merco1lem12 1693 . . 3 ((𝜑 → ((𝜑𝜓) → 𝜓)) → ((((𝜒𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → 𝜓)))
97, 8ax-mp 5 . 2 ((((𝜒𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → 𝜓))
10 merco1 1678 . 2 (((((𝜒𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → 𝜓)) → ((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒)))
119, 10ax-mp 5 1 ((((𝜑𝜓) → 𝜓) → 𝜒) → (𝜑𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wfal 1528
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 1526  df-fal 1529
This theorem is referenced by:  merco1lem15  1696  retbwax1  1700
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