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Theorem elab2 2684
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 (A Bψ)
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 2683 . 2 (A V → (A Bψ))
51, 4ax-mp 7 1 (A Bψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  {cab 2023  Vcvv 2551
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553
This theorem is referenced by:  elpw  3357  elint  3612  opabid  3985  elrn2  4519  elimasn  4635  oprabid  5480  tfrlem3a  5866  addnqprlemrl  6538  addnqprlemru  6539  addnqprlemfl  6540  addnqprlemfu  6541  archpr  6615  cauappcvgprlemladdfu  6626  cauappcvgprlemladdfl  6627  caucvgprlemladdfu  6648
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