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Mirrors > Home > MPE Home > Th. List > consensus | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
consensus | ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
2 | orc 866 | . . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
3 | 2 | adantrr 717 | . . . 4 ⊢ ((𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
4 | olc 867 | . . . . 5 ⊢ ((¬ 𝜑 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
5 | 4 | adantrl 716 | . . . 4 ⊢ ((¬ 𝜑 ∧ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
6 | 3, 5 | pm2.61ian 812 | . . 3 ⊢ ((𝜓 ∧ 𝜒) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
7 | 1, 6 | jaoi 856 | . 2 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) → ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
8 | orc 866 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) → (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒))) | |
9 | 7, 8 | impbii 212 | 1 ⊢ ((((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)) ∨ (𝜓 ∧ 𝜒)) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∨ wo 846 |
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 210 df-an 400 df-or 847 |
This theorem is referenced by: (None) |
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