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Mirrors > Home > MPE Home > Th. List > df-cad | Structured version Visualization version GIF version |
Description: Definition of the "carry" output of the full adder. It is true when at least two arguments are true, so it is equal to the "majority" function on three variables. See cador 1609 and cadan 1610 for alternate definitions. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
df-cad | ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | wps | . . 3 wff 𝜓 | |
3 | wch | . . 3 wff 𝜒 | |
4 | 1, 2, 3 | wcad 1607 | . 2 wff cadd(𝜑, 𝜓, 𝜒) |
5 | 1, 2 | wa 396 | . . 3 wff (𝜑 ∧ 𝜓) |
6 | 1, 2 | wxo 1509 | . . . 4 wff (𝜑 ⊻ 𝜓) |
7 | 3, 6 | wa 396 | . . 3 wff (𝜒 ∧ (𝜑 ⊻ 𝜓)) |
8 | 5, 7 | wo 845 | . 2 wff ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) |
9 | 4, 8 | wb 205 | 1 wff (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) |
Colors of variables: wff setvar class |
This definition is referenced by: cador 1609 cadbi123d 1611 cadcoma 1613 cad11 1617 cad0 1619 cad0OLD 1620 |
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