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Theorem riota2 7385
Description: This theorem shows a condition that allows to represent a descriptor with a class expression 𝐵. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 10-Dec-2016.)
Hypothesis
Ref Expression
riota2.1 (𝑥 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
riota2 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem riota2
StepHypRef Expression
1 nfcv 2904 . 2 𝑥𝐵
2 nfv 1918 . 2 𝑥𝜓
3 riota2.1 . 2 (𝑥 = 𝐵 → (𝜑𝜓))
41, 2, 3riota2f 7384 1 ((𝐵𝐴 ∧ ∃!𝑥𝐴 𝜑) → (𝜓 ↔ (𝑥𝐴 𝜑) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ∃!wreu 3375  crio 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-reu 3378  df-v 3477  df-un 3951  df-in 3953  df-ss 3963  df-sn 4627  df-pr 4629  df-uni 4907  df-iota 6491  df-riota 7359
This theorem is referenced by:  eqsup  9446  sup0  9456  ttrcltr  9706  fin23lem22  10317  subadd  11458  divmul  11870  fllelt  13757  flflp1  13767  flval2  13774  flbi  13776  remim  15059  resqrtcl  15195  resqrtthlem  15196  sqrtneg  15209  sqrtthlem  15304  divalgmod  16344  qnumdenbi  16675  catidd  17619  lubprop  18306  glbprop  18319  poslubd  18361  isglbd  18457  ismgmid  18579  isgrpinv  18873  pj1id  19559  coeeq  25722  scutbday  27284  eqscut  27285  scutun12  27290  scutbdaylt  27298  divsmulw  27619  ismir  27889  mireq  27895  ismidb  28008  islmib  28017  usgredg2vlem2  28462  frgrncvvdeqlem3  29533  frgr2wwlkeqm  29563  cnidOLD  29812  hilid  30391  pjpreeq  30628  cnvbraval  31340  cdj3lem2  31665  xdivmul  32068  cvmliftphtlem  34245  cvmlift3lem4  34250  cvmlift3lem6  34252  cvmlift3lem9  34255  transportprops  34943  ltflcei  36413  cmpidelt  36664  exidresid  36684  lshpkrlem1  37917  cdlemeiota  39393  dochfl1  40284  hgmapvs  40699  evlsval3  41080  fsuppind  41111  renegadd  41188  resubadd  41195  addinvcom  41247  wessf1ornlem  43814  fourierdlem50  44806
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