Theorem consensus 1050

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 864	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
3	2	adantrr 714	. . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
4		olc 865	. . . . 5 ⊢ ((¬ 𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
5	4	adantrl 713	. . . 4 ⊢ ((¬ 𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
6	3, 5	pm2.61ian 809	. . 3 ⊢ ((𝜓 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
7	1, 6	jaoi 854	. 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))
8		orc 864	. 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)))
9	7, 8	impbii 208	1 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∨ wo 844

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 206  df-an 397  df-or 845

This theorem is referenced by: (None)