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Theorem elab2 2712
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 2711 . 2 (𝐴 ∈ V → (𝐴𝐵𝜓))
51, 4ax-mp 7 1 (𝐴𝐵𝜓)
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:  elpw  3392  elint  3648  opabid  4021  elrn2  4603  elimasn  4719  oprabid  5564  tfrlem3a  5955  addnqprlemrl  6712  addnqprlemru  6713  addnqprlemfl  6714  addnqprlemfu  6715  mulnqprlemrl  6728  mulnqprlemru  6729  mulnqprlemfl  6730  mulnqprlemfu  6731  ltnqpr  6748  ltnqpri  6749  archpr  6798  cauappcvgprlemladdfu  6809  cauappcvgprlemladdfl  6810  caucvgprlemladdfu  6832  caucvgprprlemopu  6854
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