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Theorem merco1 1635
Description: A single axiom for propositional calculus offered by Meredith.

This axiom is worthy of note, due to it having only 19 symbols, not counting parentheses. The more well-known meredith 1563 has 21 symbols, sans parentheses.

See merco2 1658 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 1495 . . . . . 6 (⊥ → (¬ 𝜃 → ¬ 𝜒))
31, 2ja 173 . . . . 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 1563 . 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 1485
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 1483  df-fal 1486
This theorem is referenced by:  merco1lem1  1636  retbwax2  1638  merco1lem2  1639  merco1lem4  1641  merco1lem5  1642  merco1lem6  1643  merco1lem7  1644  merco1lem10  1648  merco1lem11  1649  merco1lem12  1650  merco1lem13  1651  merco1lem14  1652  merco1lem16  1654  merco1lem17  1655  merco1lem18  1656  retbwax1  1657
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