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Mirrors > Home > MPE Home > Th. List > cadbi123i | Structured version Visualization version GIF version |
Description: Equality theorem for the adder carry. (Contributed by Mario Carneiro, 4-Sep-2016.) |
Ref | Expression |
---|---|
cadbii.1 | ⊢ (𝜑 ↔ 𝜓) |
cadbii.2 | ⊢ (𝜒 ↔ 𝜃) |
cadbii.3 | ⊢ (𝜏 ↔ 𝜂) |
Ref | Expression |
---|---|
cadbi123i | ⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cadbii.1 | . . . 4 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → (𝜑 ↔ 𝜓)) |
3 | cadbii.2 | . . . 4 ⊢ (𝜒 ↔ 𝜃) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → (𝜒 ↔ 𝜃)) |
5 | cadbii.3 | . . . 4 ⊢ (𝜏 ↔ 𝜂) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → (𝜏 ↔ 𝜂)) |
7 | 2, 4, 6 | cadbi123d 1616 | . 2 ⊢ (⊤ → (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂))) |
8 | 7 | mptru 1549 | 1 ⊢ (cadd(𝜑, 𝜒, 𝜏) ↔ cadd(𝜓, 𝜃, 𝜂)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ⊤wtru 1543 caddwcad 1612 |
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 397 df-or 845 df-xor 1507 df-tru 1545 df-cad 1613 |
This theorem is referenced by: (None) |
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