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Theorem elrab3 2895
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 2894 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
32baib 919 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  {crab 2459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2740
This theorem is referenced by:  unimax  3844  undifexmid  4194  frind  4353  ordtriexmidlem2  4520  ordtriexmid  4521  ontriexmidim  4522  ordtri2orexmid  4523  onsucelsucexmid  4530  0elsucexmid  4565  ordpwsucexmid  4570  ordtri2or2exmid  4571  ontri2orexmidim  4572  canth  5829  acexmidlema  5866  acexmidlemb  5867  isnumi  7181  genpelvl  7511  genpelvu  7512  cauappcvgprlemladdru  7655  cauappcvgprlem1  7658  caucvgprlem1  7678  sup3exmid  8914  supinfneg  9595  infsupneg  9596  supminfex  9597  ublbneg  9613  negm  9615  hashinfuni  10757  infssuzex  11950  gcddvds  11964  dvdslegcd  11965  bezoutlemsup  12010  uzwodc  12038  lcmval  12063  dvdslcm  12069  isprm2lem  12116
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