Theorem consensus 1058

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 22	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
2		orc 873	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
3	2	adantrr 723	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
4		olc 874	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
5	4	adantrl 722	. . . 4 ⊢ ((¬ 𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
6	3, 5	pm2.61ian 817	. . 3 ⊢ ((𝜓 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
7	1, 6	jaoi 863	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
8		orc 873	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
9	7, 8	impbii 210	1 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853

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 208  df-an 397  df-or 854

This theorem is referenced by: (None)