Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cad1 | Structured version Visualization version GIF version |
Description: If one input is true, then the adder carry is true exactly when at least one of the other two inputs is true. (Contributed by Mario Carneiro, 8-Sep-2016.) (Proof shortened by Wolf Lammen, 19-Jun-2020.) |
Ref | Expression |
---|---|
cad1 | ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | olc 864 | . . . 4 ⊢ (𝜒 → (𝜑 ∨ 𝜒)) | |
2 | olc 864 | . . . 4 ⊢ (𝜒 → (𝜓 ∨ 𝜒)) | |
3 | 1, 2 | jca 514 | . . 3 ⊢ (𝜒 → ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) |
4 | 3 | biantrud 534 | . 2 ⊢ (𝜒 → ((𝜑 ∨ 𝜓) ↔ ((𝜑 ∨ 𝜓) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))))) |
5 | cadan 1610 | . . 3 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒))) | |
6 | 3anass 1091 | . . 3 ⊢ (((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)) ↔ ((𝜑 ∨ 𝜓) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))) | |
7 | 5, 6 | bitri 277 | . 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∨ 𝜓) ∧ ((𝜑 ∨ 𝜒) ∧ (𝜓 ∨ 𝜒)))) |
8 | 4, 7 | syl6rbbr 292 | 1 ⊢ (𝜒 → (cadd(𝜑, 𝜓, 𝜒) ↔ (𝜑 ∨ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 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 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-xor 1502 df-cad 1608 |
This theorem is referenced by: cadifp 1619 sadadd2lem2 15801 sadcaddlem 15808 |
Copyright terms: Public domain | W3C validator |