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Theorem deceq2 11330
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 6531 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2 df-dec 11322 . 2 𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3 df-dec 11322 . 2 𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
41, 2, 33eqtr4g 2664 1 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  (class class class)co 6523  1c1 9789   + caddc 9791   · cmul 9793  9c9 10920  cdc 11321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-rex 2897  df-rab 2900  df-v 3170  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-iota 5750  df-fv 5794  df-ov 6526  df-dec 11322
This theorem is referenced by:  deceq2i  11333
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