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Theorem opeq1d 4381
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 4375 . 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 1480  cop 4159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160
This theorem is referenced by:  oteq1  4384  oteq2  4385  opth  4910  elsnxp  5639  cbvoprab2  6682  unxpdomlem1  8109  mulcanenq  9727  ax1rid  9927  axrnegex  9928  fseq1m1p1  12353  uzrdglem  12693  swrd0swrd  13394  swrdccat  13425  swrdccat3a  13426  swrdccat3blem  13427  cshw0  13472  cshwmodn  13473  s2prop  13583  s4prop  13586  fsum2dlem  14424  fprod2dlem  14630  ruclem1  14880  imasaddvallem  16105  iscatd2  16258  moni  16312  homadmcd  16608  curf1  16781  curf1cl  16784  curf2  16785  hofcl  16815  gsum2dlem2  18286  imasdsf1olem  22083  ovoliunlem1  23172  cxpcn3  24384  axlowdimlem15  25731  axlowdim  25736  nvi  27309  nvop  27371  phop  27513  br8d  29256  fgreu  29305  1stpreimas  29317  smatfval  29635  smatrcl  29636  smatlem  29637  fvproj  29673  mvhfval  31130  mpst123  31137  br8  31345  fvtransport  31754  rfovcnvf1od  37747
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