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Theorem consensus 925
 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 ((((φ ψ) φ χ)) (ψ χ)) ↔ ((φ ψ) φ χ)))

Proof of Theorem consensus
StepHypRef Expression
1 id 19 . . 3 (((φ ψ) φ χ)) → ((φ ψ) φ χ)))
2 orc 374 . . . . 5 ((φ ψ) → ((φ ψ) φ χ)))
32adantrr 697 . . . 4 ((φ (ψ χ)) → ((φ ψ) φ χ)))
4 olc 373 . . . . 5 ((¬ φ χ) → ((φ ψ) φ χ)))
54adantrl 696 . . . 4 ((¬ φ (ψ χ)) → ((φ ψ) φ χ)))
63, 5pm2.61ian 765 . . 3 ((ψ χ) → ((φ ψ) φ χ)))
71, 6jaoi 368 . 2 ((((φ ψ) φ χ)) (ψ χ)) → ((φ ψ) φ χ)))
8 orc 374 . 2 (((φ ψ) φ χ)) → (((φ ψ) φ χ)) (ψ χ)))
97, 8impbii 180 1 ((((φ ψ) φ χ)) (ψ χ)) ↔ ((φ ψ) φ χ)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360 This theorem is referenced by: (None)
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