Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  deceq2 GIF version

Theorem deceq2 9194
 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 5782 . 2 (𝐴 = 𝐵 → (((9 + 1) · 𝐶) + 𝐴) = (((9 + 1) · 𝐶) + 𝐵))
2 df-dec 9190 . 2 𝐶𝐴 = (((9 + 1) · 𝐶) + 𝐴)
3 df-dec 9190 . 2 𝐶𝐵 = (((9 + 1) · 𝐶) + 𝐵)
41, 2, 33eqtr4g 2197 1 (𝐴 = 𝐵𝐶𝐴 = 𝐶𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  (class class class)co 5774  1c1 7628   + caddc 7630   · cmul 7632  9c9 8785  ;cdc 9189 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-iota 5088  df-fv 5131  df-ov 5777  df-dec 9190 This theorem is referenced by:  deceq2i  9196
 Copyright terms: Public domain W3C validator