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Theorem elab2 2690
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 2689 . 2  |-  ( A  e.  _V  ->  ( A  e.  B  <->  ps )
)
51, 4ax-mp 7 1  |-  ( A  e.  B  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   {cab 2026   _Vcvv 2557
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559
This theorem is referenced by:  elpw  3365  elint  3621  opabid  3994  elrn2  4576  elimasn  4692  oprabid  5537  tfrlem3a  5925  addnqprlemrl  6653  addnqprlemru  6654  addnqprlemfl  6655  addnqprlemfu  6656  mulnqprlemrl  6669  mulnqprlemru  6670  mulnqprlemfl  6671  mulnqprlemfu  6672  ltnqpr  6689  ltnqpri  6690  archpr  6739  cauappcvgprlemladdfu  6750  cauappcvgprlemladdfl  6751  caucvgprlemladdfu  6773  caucvgprprlemopu  6795
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