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Theorem elab 2802
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 1-Aug-1994.)
Hypotheses
Ref Expression
elab.1 𝐴 ∈ V
elab.2 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
elab (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab
StepHypRef Expression
1 nfv 1493 . 2 𝑥𝜓
2 elab.1 . 2 𝐴 ∈ V
3 elab.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabf 2801 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  {cab 2103  Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662
This theorem is referenced by:  ralab  2817  rexab  2819  intab  3770  dfiin2g  3816  dfiunv2  3819  uniuni  4342  dcextest  4465  peano5  4482  finds  4484  finds2  4485  funcnvuni  5162  fun11iun  5356  elabrex  5627  abrexco  5628  mapval2  6540  ssenen  6713  snexxph  6806  sbthlem2  6814  indpi  7118  nqprm  7318  nqprrnd  7319  nqprdisj  7320  nqprloc  7321  nqprl  7327  nqpru  7328  cauappcvgprlem2  7436  caucvgprlem2  7456  peano1nnnn  7628  peano2nnnn  7629  1nn  8695  peano2nn  8696  dfuzi  9119  hashfacen  10534  shftfvalg  10545  ovshftex  10546  shftfval  10548  txdis1cn  12358  bj-ssom  13030
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