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Mirrors > Home > ILE Home > Th. List > deceq1 | GIF version |
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
deceq1 | ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5736 | . . 3 ⊢ (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵)) | |
2 | 1 | oveq1d 5743 | . 2 ⊢ (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶)) |
3 | df-dec 9087 | . 2 ⊢ ;𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶) | |
4 | df-dec 9087 | . 2 ⊢ ;𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶) | |
5 | 2, 3, 4 | 3eqtr4g 2172 | 1 ⊢ (𝐴 = 𝐵 → ;𝐴𝐶 = ;𝐵𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 (class class class)co 5728 1c1 7548 + caddc 7550 · cmul 7552 9c9 8688 ;cdc 9086 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-rex 2396 df-v 2659 df-un 3041 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-iota 5046 df-fv 5089 df-ov 5731 df-dec 9087 |
This theorem is referenced by: deceq1i 9092 |
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