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Mirrors > Home > MPE Home > Th. List > cad0 | Structured version Visualization version GIF version |
Description: If one input is false, then the adder carry is true exactly when both of the other two inputs are true. (Contributed by Mario Carneiro, 8-Sep-2016.) |
Ref | Expression |
---|---|
cad0 | ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cad 1537 | . 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | |
2 | idd 24 | . . . 4 ⊢ (¬ 𝜒 → ((𝜑 ∧ 𝜓) → (𝜑 ∧ 𝜓))) | |
3 | pm2.21 119 | . . . . 5 ⊢ (¬ 𝜒 → (𝜒 → (𝜑 ∧ 𝜓))) | |
4 | 3 | adantrd 483 | . . . 4 ⊢ (¬ 𝜒 → ((𝜒 ∧ (𝜑 ⊻ 𝜓)) → (𝜑 ∧ 𝜓))) |
5 | 2, 4 | jaod 394 | . . 3 ⊢ (¬ 𝜒 → (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) → (𝜑 ∧ 𝜓))) |
6 | orc 399 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓)))) | |
7 | 5, 6 | impbid1 214 | . 2 ⊢ (¬ 𝜒 → (((𝜑 ∧ 𝜓) ∨ (𝜒 ∧ (𝜑 ⊻ 𝜓))) ↔ (𝜑 ∧ 𝜓))) |
8 | 1, 7 | syl5bb 271 | 1 ⊢ (¬ 𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ⊻ wxo 1456 caddwcad 1536 |
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 196 df-or 384 df-an 385 df-cad 1537 |
This theorem is referenced by: cadifp 1548 sadadd2lem2 15010 sadcaddlem 15017 saddisjlem 15024 |
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