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Theorem elab2 2761
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2.1 𝐴 ∈ V
elab2.2 (𝑥 = 𝐴 → (𝜑𝜓))
elab2.3 𝐵 = {𝑥𝜑}
Assertion
Ref Expression
elab2 (𝐴𝐵𝜓)
Distinct variable groups:   𝜓,𝑥   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab2
StepHypRef Expression
1 elab2.1 . 2 𝐴 ∈ V
2 elab2.2 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
3 elab2.3 . . 3 𝐵 = {𝑥𝜑}
42, 3elab2g 2760 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4ax-mp 7 1 (𝐴𝐵𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1289  wcel 1438  {cab 2074  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  elpw  3431  elint  3689  opabid  4075  elrn2  4665  elimasn  4786  oprabid  5663  tfrlem3a  6057  tfrcllemsucaccv  6101  tfrcllembxssdm  6103  tfrcllemres  6109  addnqprlemrl  7095  addnqprlemru  7096  addnqprlemfl  7097  addnqprlemfu  7098  mulnqprlemrl  7111  mulnqprlemru  7112  mulnqprlemfl  7113  mulnqprlemfu  7114  ltnqpr  7131  ltnqpri  7132  archpr  7181  cauappcvgprlemladdfu  7192  cauappcvgprlemladdfl  7193  caucvgprlemladdfu  7215  caucvgprprlemopu  7237
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