MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1lem10 Structured version   Visualization version   GIF version

Theorem merco1lem10 1691
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
merco1lem10 (((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑))

Proof of Theorem merco1lem10
StepHypRef Expression
1 merco1 1678 . . 3 (((((𝜒𝜑) → (𝜏 → ⊥)) → 𝜑) → (𝜑𝜓)) → (((𝜑𝜓) → 𝜒) → (𝜏𝜒)))
2 merco1lem2 1682 . . 3 ((((((𝜒𝜑) → (𝜏 → ⊥)) → 𝜑) → (𝜑𝜓)) → (((𝜑𝜓) → 𝜒) → (𝜏𝜒))) → ((((𝜑𝜓) → (𝜃 → ⊥)) → ((((𝜒𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑𝜓) → 𝜒) → (𝜏𝜒))))
31, 2ax-mp 5 . 2 ((((𝜑𝜓) → (𝜃 → ⊥)) → ((((𝜒𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑𝜓) → 𝜒) → (𝜏𝜒)))
4 merco1 1678 . 2 (((((𝜑𝜓) → (𝜃 → ⊥)) → ((((𝜒𝜑) → (𝜏 → ⊥)) → 𝜑) → ⊥)) → (((𝜑𝜓) → 𝜒) → (𝜏𝜒))) → (((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑)))
53, 4ax-mp 5 1 (((((𝜑𝜓) → 𝜒) → (𝜏𝜒)) → 𝜑) → (𝜃𝜑))
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:  retbwax1  1700
  Copyright terms: Public domain W3C validator