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Theorem consensus 998
 Description: The consensus theorem. This theorem and its dual (with ∨ and ∧ interchanged) are commonly used in computer logic design to eliminate redundant terms from Boolean expressions. Specifically, we prove that the term (𝜓 ∧ 𝜒) on the left-hand side is redundant. (Contributed by NM, 16-May-2003.) (Proof shortened by Andrew Salmon, 13-May-2011.) (Proof shortened by Wolf Lammen, 20-Jan-2013.)
Assertion
Ref Expression
consensus ((((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))

Proof of Theorem consensus
StepHypRef Expression
1 id 22 . . 3 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
2 orc 400 . . . . 5 ((𝜑𝜓) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
32adantrr 752 . . . 4 ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
4 olc 399 . . . . 5 ((¬ 𝜑𝜒) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
54adantrl 751 . . . 4 ((¬ 𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
63, 5pm2.61ian 830 . . 3 ((𝜓𝜒) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
71, 6jaoi 394 . 2 ((((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
8 orc 400 . 2 (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) → (((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)))
97, 8impbii 199 1 ((((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 383   ∧ wa 384 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 385  df-an 386 This theorem is referenced by: (None)
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