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Theorem elab2 3077
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2.1  |-  A  e. 
_V
elab2.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab2.3  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab2  |-  ( A  e.  B  <->  ps )
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)

Proof of Theorem elab2
StepHypRef Expression
1 elab2.1 . 2  |-  A  e. 
_V
2 elab2.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 elab2.3 . . 3  |-  B  =  { x  |  ph }
42, 3elab2g 3076 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4ax-mp 8 1  |-  ( A  e.  B  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948
This theorem is referenced by:  elpw  3797  elint  4048  opabid  4453  elrn2  5101  elimasn  5221  oprabid  6097  cardprclem  7858  iunfictbso  7987  aceq3lem  7993  dfac5lem4  7999  kmlem9  8030  domtriomlem  8314  ltexprlem3  8907  ltexprlem4  8908  reclem2pr  8917  reclem3pr  8918  supsrlem  8978  supmul1  9965  supmullem1  9966  supmullem2  9967  supmul  9968  sqrlem6  12045  infcvgaux2i  12629  mertenslem1  12653  mertenslem2  12654  4sqlem12  13316  conjnmzb  15032  sylow3lem2  15254  txuni2  17589  xkoopn  17613  met2ndci  18544  2sqlem8  21148  2sqlem11  21151  subfacp1lem3  24860  subfacp1lem5  24862  soseq  25521  nofulllem5  25653  supaddc  26228  supadd  26229  heiborlem1  26511  heiborlem6  26516  heiborlem8  26518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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