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Mirrors > Home > MPE Home > Th. List > cadcomb | Structured version Visualization version GIF version |
Description: Commutative law for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) (Proof shortened by Wolf Lammen, 11-Jul-2020.) |
Ref | Expression |
---|---|
cadcomb | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cadan 1610 | . . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | |
2 | 3ancoma 1098 | . . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜓 ∨ 𝜒))) | |
3 | orcom 868 | . . . 4 ⊢ ((𝜓 ∨ 𝜒) ↔ (𝜒 ∨ 𝜓)) | |
4 | 3 | 3anbi3i 1159 | . . 3 ⊢ (((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓))) |
5 | 1, 2, 4 | 3bitri 296 | . 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓))) |
6 | cadan 1610 | . 2 ⊢ (cadd(𝜑, 𝜒, 𝜓) ↔ ((𝜑 ∨ 𝜒) ∧ (𝜑 ∨ 𝜓) ∧ (𝜒 ∨ 𝜓))) | |
7 | 5, 6 | bitr4i 277 | 1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜑, 𝜒, 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∨ wo 845 ∧ w3a 1087 caddwcad 1607 |
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 846 df-3or 1088 df-3an 1089 df-xor 1510 df-cad 1608 |
This theorem is referenced by: cadrot 1615 |
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