Theorem dn1 1000

Description: A single axiom for Boolean algebra known as DN₁. See McCune, Veroff, Fitelson, Harris, Feist, Wos, Short single axioms for Boolean algebra, Journal of Automated Reasoning, 29(1):1--16, 2002. (https://www.cs.unm.edu/~mccune/papers/basax/v12.pdf). (Contributed by Jeff Hankins, 3-Jul-2009.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 6-Jan-2013.)

Assertion

Ref	Expression
dn1	⊢ (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒)

Proof of Theorem dn1

Step	Hyp	Ref	Expression
1		pm2.45 411	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → ¬ 𝜑)
2		imnan 437	. . . . 5 ⊢ ((¬ (𝜑 ∨ 𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑))
3	1, 2	mpbi 219	. . . 4 ⊢ ¬ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)
4	3	biorfi 421	. . 3 ⊢ (𝜒 ↔ (𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)))
5		orcom 401	. . . 4 ⊢ ((𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒))
6		ordir 905	. . . 4 ⊢ (((¬ (𝜑 ∨ 𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
7	5, 6	bitri 263	. . 3 ⊢ ((𝜒 ∨ (¬ (𝜑 ∨ 𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
8	4, 7	bitri 263	. 2 ⊢ (𝜒 ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)))
9		pm4.45 720	. . . . 5 ⊢ (𝜒 ↔ (𝜒 ∧ (𝜒 ∨ 𝜃)))
10		anor 509	. . . . 5 ⊢ ((𝜒 ∧ (𝜒 ∨ 𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
11	9, 10	bitri 263	. . . 4 ⊢ (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))
12	11	orbi2i 540	. . 3 ⊢ ((𝜑 ∨ 𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃))))
13	12	anbi2i 726	. 2 ⊢ (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ 𝜒)) ↔ ((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
14		anor 509	. 2 ⊢ (((¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ ¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))))
15	8, 13, 14	3bitrri 286	1 ⊢ (¬ (¬ (¬ (𝜑 ∨ 𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒 ∨ 𝜃)))) ↔ 𝜒)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by: (None)