Theorem meran3 36414

Description: A single axiom for propositional calculus discovered by C. A. Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)

Assertion

Ref	Expression
meran3	⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))

Proof of Theorem meran3

Step	Hyp	Ref	Expression
1		pm2.3 925	. . . . . 6 ⊢ ((𝜒 ∨ (𝜃 ∨ 𝜏)) → (𝜒 ∨ (𝜏 ∨ 𝜃)))
2	1	imim2i 16	. . . . 5 ⊢ (((¬ 𝜑 ∨ 𝜓) → (𝜒 ∨ (𝜃 ∨ 𝜏))) → ((¬ 𝜑 ∨ 𝜓) → (𝜒 ∨ (𝜏 ∨ 𝜃))))
3		pm1.5 920	. . . . 5 ⊢ ((𝜒 ∨ (𝜏 ∨ 𝜃)) → (𝜏 ∨ (𝜒 ∨ 𝜃)))
4	2, 3	syl6 35	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) → (𝜒 ∨ (𝜃 ∨ 𝜏))) → ((¬ 𝜑 ∨ 𝜓) → (𝜏 ∨ (𝜒 ∨ 𝜃))))
5		imor 854	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) → (𝜒 ∨ (𝜃 ∨ 𝜏))) ↔ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))))
6		imor 854	. . . 4 ⊢ (((¬ 𝜑 ∨ 𝜓) → (𝜏 ∨ (𝜒 ∨ 𝜃))) ↔ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜏 ∨ (𝜒 ∨ 𝜃))))
7	4, 5, 6	3imtr3i 291	. . 3 ⊢ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) → (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜏 ∨ (𝜒 ∨ 𝜃))))
8		meran1 36412	. . . 4 ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜏 ∨ (𝜒 ∨ 𝜃))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))
9	8	imorri 856	. . 3 ⊢ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜏 ∨ (𝜒 ∨ 𝜃))) → (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))
10	7, 9	syl 17	. 2 ⊢ ((¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) → (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))
11	10	imori 855	1 ⊢ (¬ (¬ (¬ 𝜑 ∨ 𝜓) ∨ (𝜒 ∨ (𝜃 ∨ 𝜏))) ∨ (¬ (¬ 𝜒 ∨ 𝜑) ∨ (𝜏 ∨ (𝜃 ∨ 𝜑))))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848

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-or 849

This theorem is referenced by: (None)