Theorem merco1 1707

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 1635 has 21 symbols, sans parentheses.

See merco2 1730 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	⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑)))

Proof of Theorem merco1

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . 6 ⊢ (¬ 𝜒 → (¬ 𝜃 → ¬ 𝜒))
2		falim 1550	. . . . . 6 ⊢ (⊥ → (¬ 𝜃 → ¬ 𝜒))
3	1, 2	ja 186	. . . . 5 ⊢ ((𝜒 → ⊥) → (¬ 𝜃 → ¬ 𝜒))
4	3	imim2i 16	. . . 4 ⊢ (((𝜑 → 𝜓) → (𝜒 → ⊥)) → ((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)))
5	4	imim1i 63	. . 3 ⊢ ((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → (((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃))
6	5	imim1i 63	. 2 ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏))
7		meredith 1635	. 2 ⊢ (((((𝜑 → 𝜓) → (¬ 𝜃 → ¬ 𝜒)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑)))
8	6, 7	syl 17	1 ⊢ (((((𝜑 → 𝜓) → (𝜒 → ⊥)) → 𝜃) → 𝜏) → ((𝜏 → 𝜑) → (𝜒 → 𝜑)))

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1545

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 206  df-tru 1536  df-fal 1546

This theorem is referenced by:  merco1lem1  1708  retbwax2  1710  merco1lem2  1711  merco1lem4  1713  merco1lem5  1714  merco1lem6  1715  merco1lem7  1716  merco1lem10  1720  merco1lem11  1721  merco1lem12  1722  merco1lem13  1723  merco1lem14  1724  merco1lem16  1726  merco1lem17  1727  merco1lem18  1728  retbwax1  1729