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Theorem elab2 2917
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 2916 . 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 176    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788
This theorem is referenced by:  elpw  3631  elint  3868  opabid  4271  elrn2  4918  elimasn  5038  oprabid  5882  cardprclem  7612  iunfictbso  7741  aceq3lem  7747  dfac5lem4  7753  kmlem9  7784  domtriomlem  8068  ltexprlem3  8662  ltexprlem4  8663  reclem2pr  8672  reclem3pr  8673  supsrlem  8733  supmul1  9719  supmullem1  9720  supmullem2  9721  supmul  9722  sqrlem6  11733  infcvgaux2i  12316  mertenslem1  12340  mertenslem2  12341  4sqlem12  13003  conjnmzb  14717  sylow3lem2  14939  txuni2  17260  xkoopn  17284  met2ndci  18068  2sqlem8  20611  2sqlem11  20614  subfacp1lem3  23713  subfacp1lem5  23715  soseq  24254  nofulllem5  24360  fisub  25554  heiborlem1  26535  heiborlem6  26540  heiborlem8  26542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790
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