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Mirrors > Home > MPE Home > Th. List > cad11 | Structured version Visualization version GIF version |
Description: If (at least) two inputs are true, then the adder carry is true. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
cad11 | ⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orc 400 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | |
2 | df-cad 1546 | . 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | |
3 | 1, 2 | sylibr 224 | 1 ⊢ ((𝜑 ∧ 𝜓) → cadd(𝜑, 𝜓, 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 ⊻ wxo 1464 caddwcad 1545 |
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 197 df-or 385 df-cad 1546 |
This theorem is referenced by: cadtru 1559 |
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