MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elabg Unicode version

Theorem elabg 2915
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 2419 . 2  |-  F/_ x A
2 nfv 1605 . 2  |-  F/ x ps
3 elabg.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
41, 2, 3elabgf 2912 1  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684   {cab 2269
This theorem is referenced by:  elab2g  2916  intmin3  3890  finds  4682  elxpi  4705  elfi  7167  inficl  7178  dffi3  7184  scott0  7556  elgch  8244  nqpr  8638  hashf1lem1  11393  efgcpbllemb  15064  frgpuplem  15081  lspsn  15759  eltg  16695  eltg2  16696  fbssfi  17532  mpfind  19428  pf1ind  19438  elabreximd  23039  ballotlemfmpn  23053  abfmpunirn  23216  orvcval  23658  eloi  25086  elixp2b  25154  domrancur1b  25200  domrancur1c  25202  isoriso2  25213  isdir2  25292  intopcoaconlem3b  25538  iscol2  26093  islocfin  26296  setindtrs  27118  rngunsnply  27378  afvelrnb  28025  afvelrnb0  28026  islshpkrN  29310
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
  Copyright terms: Public domain W3C validator