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Theorem elabf 3245
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 2619 . . 3  |-  F/_ x A
3 elabf.1 . . 3  |-  F/ x ps
4 elabf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
52, 3, 4elabgf 3244 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
61, 5ax-mp 5 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395   F/wnf 1617    e. wcel 1819   {cab 2442   _Vcvv 3109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111
This theorem is referenced by:  elab  3246  dfon2lem1  29432  sdclem2  30440  sdclem1  30441
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