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Theorem elrab3 2763
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)
Hypothesis
Ref Expression
elrab.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elrab3 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab3
StepHypRef Expression
1 elrab.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
21elrab 2762 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
32baib 864 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wcel 1436  {crab 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rab 2364  df-v 2617
This theorem is referenced by:  unimax  3672  undifexmid  4004  frind  4155  ordtriexmidlem2  4312  ordtriexmid  4313  ordtri2orexmid  4314  onsucelsucexmid  4321  0elsucexmid  4356  ordpwsucexmid  4361  ordtri2or2exmid  4362  acexmidlema  5606  acexmidlemb  5607  isnumi  6757  genpelvl  7018  genpelvu  7019  cauappcvgprlemladdru  7162  cauappcvgprlem1  7165  caucvgprlem1  7185  supinfneg  9018  infsupneg  9019  supminfex  9020  ublbneg  9033  negm  9035  hashinfuni  10103  infssuzex  10870  gcddvds  10880  dvdslegcd  10881  bezoutlemsup  10923  lcmval  10970  dvdslcm  10976  isprm2lem  11023
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