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Theorem dn1 1054
 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 pm2.45 879 . . . . 5 (¬ (𝜑𝜓) → ¬ 𝜑)
2 imnan 403 . . . . 5 ((¬ (𝜑𝜓) → ¬ 𝜑) ↔ ¬ (¬ (𝜑𝜓) ∧ 𝜑))
31, 2mpbi 233 . . . 4 ¬ (¬ (𝜑𝜓) ∧ 𝜑)
43biorfi 936 . . 3 (𝜒 ↔ (𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)))
5 orcom 867 . . . 4 ((𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒))
6 ordir 1004 . . . 4 (((¬ (𝜑𝜓) ∧ 𝜑) ∨ 𝜒) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
75, 6bitri 278 . . 3 ((𝜒 ∨ (¬ (𝜑𝜓) ∧ 𝜑)) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
84, 7bitri 278 . 2 (𝜒 ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)))
9 pm4.45 995 . . . . 5 (𝜒 ↔ (𝜒 ∧ (𝜒𝜃)))
10 anor 980 . . . . 5 ((𝜒 ∧ (𝜒𝜃)) ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
119, 10bitri 278 . . . 4 (𝜒 ↔ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))
1211orbi2i 910 . . 3 ((𝜑𝜒) ↔ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃))))
1312anbi2i 625 . 2 (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑𝜒)) ↔ ((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
14 anor 980 . 2 (((¬ (𝜑𝜓) ∨ 𝜒) ∧ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ ¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))))
158, 13, 143bitrri 301 1 (¬ (¬ (¬ (𝜑𝜓) ∨ 𝜒) ∨ ¬ (𝜑 ∨ ¬ (¬ 𝜒 ∨ ¬ (𝜒𝜃)))) ↔ 𝜒)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844 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 210  df-an 400  df-or 845 This theorem is referenced by: (None)
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