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Theorem consensus 1039
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 (𝜓𝜒) 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 848 . . . . 5 ((𝜑𝜓) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
32adantrr 690 . . . 4 ((𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
4 olc 849 . . . . 5 ((¬ 𝜑𝜒) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
54adantrl 689 . . . 4 ((¬ 𝜑 ∧ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
63, 5pm2.61ian 806 . . 3 ((𝜓𝜒) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
71, 6jaoi 838 . 2 ((((𝜑𝜓) ∨ (¬ 𝜑𝜒)) ∨ (𝜓𝜒)) → ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
8 orc 848 . 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  wa 382  wo 828
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-an 383  df-or 829
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator