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Theorem opabbidv 4648
 Description: Equivalent wff's yield equal ordered-pair class abstractions (deduction rule). (Contributed by NM, 15-May-1995.)
Hypothesis
Ref Expression
opabbidv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
opabbidv (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem opabbidv
StepHypRef Expression
1 nfv 1830 . 2 𝑥𝜑
2 nfv 1830 . 2 𝑦𝜑
3 opabbidv.1 . 2 (𝜑 → (𝜓𝜒))
41, 2, 3opabbid 4647 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ 𝜒})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  {copab 4642 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-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-opab 4644 This theorem is referenced by:  opabbii  4649  csbopab  4932  csbopabgALT  4933  csbmpt12  4934  xpeq1  5052  xpeq2  5053  opabbi2dv  5193  csbcnvgALT  5229  resopab2  5368  mptcnv  5453  cores  5555  xpco  5592  dffn5  6151  f1oiso2  6502  fvmptopab1  6594  f1ocnvd  6782  ofreq  6798  bropopvvv  7142  bropfvvvv  7144  fnwelem  7179  sprmpt2d  7237  mpt2curryd  7282  wemapwe  8477  xpcogend  13561  shftfval  13658  2shfti  13668  prdsval  15938  pwsle  15975  sectffval  16233  sectfval  16234  isfunc  16347  isfull  16393  isfth  16397  ipoval  16977  eqgfval  17465  dvdsrval  18468  dvdsrpropd  18519  ltbval  19292  opsrval  19295  lmfval  20846  xkocnv  21427  tgphaus  21730  isphtpc  22601  bcthlem1  22929  bcth  22934  ulmval  23938  lgsquadlem3  24907  iscgrg  25207  legval  25279  ishlg  25297  perpln1  25405  perpln2  25406  isperp  25407  ishpg  25451  iscgra  25501  isinag  25529  isleag  25533  wlks  26047  wlkcompim  26054  wlkelwrd  26058  wlkon  26061  trls  26066  trlon  26070  pths  26096  spths  26097  pthon  26105  spthon  26112  clwlk  26281  clwlkcompim  26292  iseupa  26492  ajfval  27048  f1o3d  28813  f1od2  28887  inftmrel  29065  isinftm  29066  metidval  29261  faeval  29636  eulerpartlemgvv  29765  eulerpart  29771  afsval  30002  cureq  32555  curf  32557  curunc  32561  fnopabeqd  32684  lcvfbr  33325  cmtfvalN  33515  cvrfval  33573  dicffval  35481  dicfval  35482  dicval  35483  dnwech  36636  aomclem8  36649  rfovcnvfvd  37321  fsovrfovd  37323  dfafn5a  39889  mptmpt2opabbrd  40335  1wlksfval  40811  wlksfval  40812  upgrtrls  40909  upgrspths1wlk  40944
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