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Theorem elabf 2913
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 2419 . . 3  |-  F/_ x A
3 elabf.1 . . 3  |-  F/ x ps
4 elabf.3 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
52, 3, 4elabgf 2912 . 2  |-  ( A  e.  _V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
61, 5ax-mp 8 1  |-  ( A  e.  { x  | 
ph }  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531    = wceq 1623    e. wcel 1684   {cab 2269   _Vcvv 2788
This theorem is referenced by:  elab  2914  dya2iocseg  23579  dfon2lem1  24139  sdclem2  26452  sdclem1  26453
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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