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Theorem opeq1d 4346
 Description: Equality deduction for ordered pairs. (Contributed by NM, 16-Dec-2006.)
Hypothesis
Ref Expression
opeq1d.1 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
opeq1d (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)

Proof of Theorem opeq1d
StepHypRef Expression
1 opeq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 opeq1 4340 . 2 (𝐴 = 𝐵 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
31, 2syl 17 1 (𝜑 → ⟨𝐴, 𝐶⟩ = ⟨𝐵, 𝐶⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  ⟨cop 4131 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-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 This theorem is referenced by:  oteq1  4349  oteq2  4350  opth  4871  elsnxp  5594  cbvoprab2  6626  unxpdomlem1  8049  mulcanenq  9661  ax1rid  9861  axrnegex  9862  fseq1m1p1  12284  uzrdglem  12618  swrd0swrd  13313  swrdccat  13344  swrdccat3a  13345  swrdccat3blem  13346  cshw0  13391  cshwmodn  13392  s2prop  13502  s4prop  13505  fsum2dlem  14343  fprod2dlem  14549  ruclem1  14799  imasaddvallem  16012  iscatd2  16165  moni  16219  homadmcd  16515  curf1  16688  curf1cl  16691  curf2  16692  hofcl  16722  gsum2dlem2  18193  imasdsf1olem  21988  ovoliunlem1  23077  cxpcn3  24289  axlowdimlem15  25636  axlowdim  25641  nvi  26853  nvop  26915  phop  27057  br8d  28802  fgreu  28854  1stpreimas  28866  smatfval  29189  smatrcl  29190  smatlem  29191  fvproj  29227  mvhfval  30684  mpst123  30691  br8  30899  fvtransport  31309  rfovcnvf1od  37318
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