MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  merco1 Structured version   Visualization version   GIF version

Theorem merco1 1713
Description: A single axiom for propositional calculus discovered by C. A. Meredith.

This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1641 has 21 symbols, sans parentheses.

See merco2 1736 for another axiom of equal length. (Contributed by Anthony Hart, 13-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion
Ref Expression
merco1 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))

Proof of Theorem merco1
StepHypRef Expression
1 ax-1 6 . . . . . 6 𝜒 → (¬ 𝜃 → ¬ 𝜒))
2 falim 1557 . . . . . 6 (⊥ → (¬ 𝜃 → ¬ 𝜒))
31, 2ja 186 . . . . 5 ((𝜒 → ⊥) → (¬ 𝜃 → ¬ 𝜒))
43imim2i 16 . . . 4 (((𝜑𝜓) → (𝜒 → ⊥)) → ((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)))
54imim1i 63 . . 3 ((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → (((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃))
65imim1i 63 . 2 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏))
7 meredith 1641 . 2 (((((𝜑𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))
86, 7syl 17 1 (((((𝜑𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏𝜑) → (𝜒𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wfal 1552
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 207  df-tru 1543  df-fal 1553
This theorem is referenced by:  merco1lem1  1714  retbwax2  1716  merco1lem2  1717  merco1lem4  1719  merco1lem5  1720  merco1lem6  1721  merco1lem7  1722  merco1lem10  1726  merco1lem11  1727  merco1lem12  1728  merco1lem13  1729  merco1lem14  1730  merco1lem16  1732  merco1lem17  1733  merco1lem18  1734  retbwax1  1735
  Copyright terms: Public domain W3C validator