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Theorem elrab3 2808
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 2807 . 2 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
32baib 885 1 (𝐴𝐵 → (𝐴 ∈ {𝑥𝐵𝜑} ↔ 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1312  wcel 1461  {crab 2392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657
This theorem is referenced by:  unimax  3734  undifexmid  4075  frind  4232  ordtriexmidlem2  4394  ordtriexmid  4395  ordtri2orexmid  4396  onsucelsucexmid  4403  0elsucexmid  4438  ordpwsucexmid  4443  ordtri2or2exmid  4444  acexmidlema  5717  acexmidlemb  5718  isnumi  6985  genpelvl  7262  genpelvu  7263  cauappcvgprlemladdru  7406  cauappcvgprlem1  7409  caucvgprlem1  7429  sup3exmid  8619  supinfneg  9286  infsupneg  9287  supminfex  9288  ublbneg  9301  negm  9303  hashinfuni  10410  infssuzex  11484  gcddvds  11494  dvdslegcd  11495  bezoutlemsup  11537  lcmval  11584  dvdslcm  11590  isprm2lem  11637
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