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Theorem elabg 3043
Description: Membership in a class abstraction, using implicit substitution. Compare Theorem 6.13 of [Quine] p. 44. (Contributed by NM, 14-Apr-1995.)
Hypothesis
Ref Expression
elabg.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
elabg  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem elabg
StepHypRef Expression
1 nfcv 2540 . 2  |-  F/_ x A
2 nfv 1626 . 2  |-  F/ x ps
3 elabg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 3040 1  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   {cab 2390
This theorem is referenced by:  elab2g  3044  intmin3  4038  finds  4830  elxpi  4853  elfi  7376  inficl  7388  dffi3  7394  scott0  7766  elgch  8453  nqpr  8847  hashf1lem1  11659  efgcpbllemb  15342  frgpuplem  15359  lspsn  16033  eltg  16977  eltg2  16978  fbssfi  17822  mpfind  19918  pf1ind  19928  elabreximd  23944  abfmpunirn  24017  orvcval  24668  islocfin  26266  setindtrs  26986  rngunsnply  27246  afvelrnb  27894  afvelrnb0  27895  islshpkrN  29603
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918
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