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

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ (((φ ∧ ψ) ∨ (¬ φ ∧ χ)) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
2		orc 374	. . . . 5 ⊢ ((φ ∧ ψ) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
3	2	adantrr 697	. . . 4 ⊢ ((φ ∧ (ψ ∧ χ)) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
4		olc 373	. . . . 5 ⊢ ((¬ φ ∧ χ) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
5	4	adantrl 696	. . . 4 ⊢ ((¬ φ ∧ (ψ ∧ χ)) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
6	3, 5	pm2.61ian 765	. . 3 ⊢ ((ψ ∧ χ) → ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))
7	1, 6	jaoi 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)))
9	7, 8	impbii 180	1 ⊢ ((((φ ∧ ψ) ∨ (¬ φ ∧ χ)) ∨ (ψ ∧ χ)) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ χ)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)