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Mirrors > Home > ILE Home > Th. List > deceq2 | GIF version |
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deceq2 | ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5790 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵)) | |
2 | df-dec 9207 | . 2 ⊢ ;𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴) | |
3 | df-dec 9207 | . 2 ⊢ ;𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵) | |
4 | 1, 2, 3 | 3eqtr4g 2198 | 1 ⊢ (𝐴 = 𝐵 → ;𝐶𝐴 = ;𝐶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 (class class class)co 5782 1c1 7645 + caddc 7647 · cmul 7649 9c9 8802 ;cdc 9206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rex 2423 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-iota 5096 df-fv 5139 df-ov 5785 df-dec 9207 |
This theorem is referenced by: deceq2i 9213 |
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