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Theorem elabf 3083
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 12-Oct-2016.)
Hypotheses
Ref Expression
elabf.1  |-  F/ x ps
elabf.2  |-  A  e. 
_V
elabf.3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabf  |-  ( A  e.  { x  | 
ph }  <->  ps )
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem elabf
StepHypRef Expression
1 elabf.2 . 2  |-  A  e. 
_V
2 nfcv 2574 . . 3  |-  F/_ x A
3 elabf.1 . . 3  |-  F/ x ps
4 elabf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
52, 3, 4elabgf 3082 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
61, 5ax-mp 5 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178   F/wnf 1554    = wceq 1653    e. wcel 1726   {cab 2424   _Vcvv 2958
This theorem is referenced by:  elab  3084  dfon2lem1  25415  sdclem2  26460  sdclem1  26461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960
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