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Theorem deceq2 12148
 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 7163 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2 df-dec 12143 . 2 𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3 df-dec 12143 . 2 𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
41, 2, 33eqtr4g 2818 1 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  (class class class)co 7155  1c1 10581   + caddc 10583   · cmul 10585  9c9 11741  ;cdc 12142 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347  df-ov 7158  df-dec 12143 This theorem is referenced by:  deceq2i  12150
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