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Theorem elrab3 2894
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 2893 . 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 2739
This theorem is referenced by:  unimax  3843  undifexmid  4193  frind  4352  ordtriexmidlem2  4519  ordtriexmid  4520  ontriexmidim  4521  ordtri2orexmid  4522  onsucelsucexmid  4529  0elsucexmid  4564  ordpwsucexmid  4569  ordtri2or2exmid  4570  ontri2orexmidim  4571  canth  5828  acexmidlema  5865  acexmidlemb  5866  isnumi  7180  genpelvl  7510  genpelvu  7511  cauappcvgprlemladdru  7654  cauappcvgprlem1  7657  caucvgprlem1  7677  sup3exmid  8913  supinfneg  9594  infsupneg  9595  supminfex  9596  ublbneg  9612  negm  9614  hashinfuni  10756  infssuzex  11949  gcddvds  11963  dvdslegcd  11964  bezoutlemsup  12009  uzwodc  12037  lcmval  12062  dvdslcm  12068  isprm2lem  12115
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