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Theorem merco1lem12 1693
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
merco1lem12 ((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓))

Proof of Theorem merco1lem12
StepHypRef Expression
1 merco1lem3 1683 . . . . 5 ((((𝜑𝜏) → (((𝜒 → (𝜑𝜏)) → 𝜑) → ⊥)) → (𝜒 → ⊥)) → (𝜒 → (𝜑𝜏)))
2 merco1 1678 . . . . 5 (((((𝜑𝜏) → (((𝜒 → (𝜑𝜏)) → 𝜑) → ⊥)) → (𝜒 → ⊥)) → (𝜒 → (𝜑𝜏))) → (((𝜒 → (𝜑𝜏)) → 𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑)))
31, 2ax-mp 5 . . . 4 (((𝜒 → (𝜑𝜏)) → 𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑))
4 merco1lem9 1690 . . . 4 ((((𝜒 → (𝜑𝜏)) → 𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑)) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑))
53, 4ax-mp 5 . . 3 (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑)
6 merco1lem11 1692 . . 3 ((((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜑) → ((((𝜓𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑))
75, 6ax-mp 5 . 2 ((((𝜓𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑)
8 merco1 1678 . 2 (((((𝜓𝜑) → (((𝜒 → (𝜑𝜏)) → 𝜑) → ⊥)) → ⊥) → 𝜑) → ((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓)))
97, 8ax-mp 5 1 ((𝜑𝜓) → (((𝜒 → (𝜑𝜏)) → 𝜑) → 𝜓))
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:  merco1lem13  1694  merco1lem14  1695
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