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Theorem deceq1 12091
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 7153 . . 3 (𝐴 = 𝐵 → ((9 + 1) · 𝐴) = ((9 + 1) · 𝐵))
21oveq1d 7160 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐴) + 𝐶) = (((9 + 1) · 𝐵) + 𝐶))
3 df-dec 12087 . 2 𝐴𝐶 = (((9 + 1) · 𝐴) + 𝐶)
4 df-dec 12087 . 2 𝐵𝐶 = (((9 + 1) · 𝐵) + 𝐶)
52, 3, 43eqtr4g 2878 1 (𝐴 = 𝐵𝐴𝐶 = 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  (class class class)co 7145  1c1 10526   + caddc 10528   · cmul 10530  9c9 11687  cdc 12086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-dec 12087
This theorem is referenced by:  deceq1i  12093
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