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Mirrors > Home > MPE Home > Th. List > cadifp | Structured version Visualization version GIF version |
Description: The value of the carry is, if the input carry is true, the disjunction, and if the input carry is false, the conjunction, of the other two inputs. (Contributed by BJ, 8-Oct-2019.) |
Ref | Expression |
---|---|
cadifp | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cad1 1616 | . 2 ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓))) | |
2 | cad0 1617 | . 2 ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) | |
3 | 1, 2 | casesifp 1075 | 1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜒, (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ∨ wo 843 if-wif 1059 caddwcad 1605 |
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 395 df-or 844 df-ifp 1060 df-3or 1086 df-3an 1087 df-xor 1508 df-cad 1606 |
This theorem is referenced by: wl-df-3mintru2 36668 |
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