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Theorem elab2 2930
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 2929 . 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 1632    e. wcel 1696   {cab 2282   _Vcvv 2801
This theorem is referenced by:  elpw  3644  elint  3884  opabid  4287  elrn2  4934  elimasn  5054  oprabid  5898  cardprclem  7628  iunfictbso  7757  aceq3lem  7763  dfac5lem4  7769  kmlem9  7800  domtriomlem  8084  ltexprlem3  8678  ltexprlem4  8679  reclem2pr  8688  reclem3pr  8689  supsrlem  8749  supmul1  9735  supmullem1  9736  supmullem2  9737  supmul  9738  sqrlem6  11749  infcvgaux2i  12332  mertenslem1  12356  mertenslem2  12357  4sqlem12  13019  conjnmzb  14733  sylow3lem2  14955  txuni2  17276  xkoopn  17300  met2ndci  18084  2sqlem8  20627  2sqlem11  20630  subfacp1lem3  23728  subfacp1lem5  23730  soseq  24325  nofulllem5  24431  supaddc  24995  supadd  24996  fisub  25657  heiborlem1  26638  heiborlem6  26643  heiborlem8  26645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803
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