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Theorem deceq1 9090
Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
Assertion
Ref Expression
deceq1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)

Proof of Theorem deceq1
StepHypRef Expression
1 oveq2 5736 . . 3 (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
21oveq1d 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)
52, 3, 43eqtr4g 2172 1 (𝐴 = 𝐵𝐴𝐶 = 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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