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Theorem elab 2710
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 1437 . 2 𝑥𝜓
2 elab.1 . 2 𝐴 ∈ V
3 elab.2 . 2 (𝑥 = 𝐴 → (𝜑𝜓))
41, 2, 3elabf 2709 1 (𝐴 ∈ {𝑥𝜑} ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  {cab 2042  Vcvv 2574
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-v 2576
This theorem is referenced by:  ralab  2724  rexab  2726  intab  3672  dfiin2g  3718  dfiunv2  3721  uniuni  4211  peano5  4349  finds  4351  finds2  4352  funcnvuni  4996  fun11iun  5175  elabrex  5425  abrexco  5426  indpi  6498  nqprm  6698  nqprrnd  6699  nqprdisj  6700  nqprloc  6701  nqprl  6707  nqpru  6708  cauappcvgprlem2  6816  caucvgprlem2  6836  peano1nnnn  6986  peano2nnnn  6987  1nn  8001  peano2nn  8002  dfuzi  8407  shftfvalg  9647  ovshftex  9648  shftfval  9650  bj-ssom  10447
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