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Theorem merco1lem13 1694
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
merco1lem13 ((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏))

Proof of Theorem merco1lem13
StepHypRef Expression
1 merco1 1678 . . . 4 (((((𝜓𝜑) → (𝜒 → ⊥)) → 𝜑) → 𝜑) → ((𝜑𝜓) → (𝜒𝜓)))
2 merco1lem4 1684 . . . 4 ((((((𝜓𝜑) → (𝜒 → ⊥)) → 𝜑) → 𝜑) → ((𝜑𝜓) → (𝜒𝜓))) → (𝜑 → ((𝜑𝜓) → (𝜒𝜓))))
31, 2ax-mp 5 . . 3 (𝜑 → ((𝜑𝜓) → (𝜒𝜓)))
4 merco1lem12 1693 . . 3 ((𝜑 → ((𝜑𝜓) → (𝜒𝜓))) → ((((𝜏𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → (𝜒𝜓))))
53, 4ax-mp 5 . 2 ((((𝜏𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → (𝜒𝜓)))
6 merco1 1678 . 2 (((((𝜏𝜑) → (𝜑 → ⊥)) → 𝜑) → ((𝜑𝜓) → (𝜒𝜓))) → ((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏)))
75, 6ax-mp 5 1 ((((𝜑𝜓) → (𝜒𝜓)) → 𝜏) → (𝜑𝜏))
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:  merco1lem14  1695  merco1lem15  1696  retbwax1  1700
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