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Theorem elrab 2721
Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 21-May-1999.)
Hypothesis
Ref Expression
elrab.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elrab (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elrab
StepHypRef Expression
1 nfcv 2194 . 2 𝑥𝐴
2 nfcv 2194 . 2 𝑥𝐵
3 nfv 1437 . 2 𝑥𝜓
4 elrab.1 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
51, 2, 3, 4elrabf 2719 1 (𝐴 ∈ {𝑥𝐵𝜑} ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  {crab 2327
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576
This theorem is referenced by:  elrab3  2722  elrab2  2723  ralrab  2725  rexrab  2727  rabsnt  3473  unimax  3642  ssintub  3661  intminss  3668  rabxfrd  4229  ordtri2or2exmidlem  4279  onsucelsucexmidlem1  4281  sefvex  5224  ssimaex  5262  acexmidlem2  5537  ssfiexmid  6367  diffitest  6375  supubti  6405  suplubti  6406  caucvgprlemladdfu  6833  caucvgprlemladdrl  6834  nnindnn  7025  nnind  8006  peano2uz2  8404  peano5uzti  8405  dfuzi  8407  uzind  8408  uzind3  8410  eluz1  8573  uzind4  8627  eqreznegel  8646  elixx1  8867  elioo2  8891  elfz1  8981  serige0  9417  expcl2lemap  9432  expclzaplem  9444  expclzap  9445  expap0i  9452  expge0  9456  expge1  9457  shftf  9659  dvdsdivcl  10162  divalgmod  10239
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