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Theorem dn1 1028
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 (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒)

Proof of Theorem dn1
StepHypRef Expression
1 pm2.45 411 . . . . 5 (¬ (𝜑𝜓) → ¬ 𝜑)
2 imnan 437 . . . . 5 ((¬ (𝜑𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑𝜓) ∧ 𝜑))
31, 2mpbi 220 . . . 4 ¬ (¬ (𝜑𝜓) ∧ 𝜑)
43biorfi 421 . . 3 (𝜒 ↔ (𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)))
5 orcom 401 . . . 4 ((𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒))
6 ordir 927 . . . 4 (((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
75, 6bitri 264 . . 3 ((𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
84, 7bitri 264 . 2 (𝜒 ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
9 pm4.45 724 . . . . 5 (𝜒 ↔ (𝜒 ∧ (𝜒𝜃)))
10 anor 509 . . . . 5 ((𝜒 ∧ (𝜒𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
119, 10bitri 264 . . . 4 (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
1211orbi2i 540 . . 3 ((𝜑𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
1312anbi2i 730 . 2 (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
14 anor 509 . 2 (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ ¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
158, 13, 143bitrri 287 1 (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  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 197  df-or 384  df-an 385
This theorem is referenced by: (None)
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