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Theorem deceq2 11378
 Description: Equality theorem for the decimal constructor. (Contributed by Mario Carneiro, 17-Apr-2015.) (Revised by AV, 6-Sep-2021.)
Assertion
Ref Expression
deceq2 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)

Proof of Theorem deceq2
StepHypRef Expression
1 oveq2 6557 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2 df-dec 11370 . 2 𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3 df-dec 11370 . 2 𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
41, 2, 33eqtr4g 2669 1 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  (class class class)co 6549  1c1 9816   + caddc 9818   · cmul 9820  9c9 10954  ;cdc 11369 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-dec 11370 This theorem is referenced by:  deceq2i  11381
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