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Theorem elab2g 3086
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 13-Sep-1995.)
Hypotheses
Ref Expression
elab2g.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
elab2g.2  |-  B  =  { x  |  ph }
Assertion
Ref Expression
elab2g  |-  ( A  e.  V  ->  ( A  e.  B  <->  ps )
)
Distinct variable groups:    ps, x    x, A
Allowed substitution hints:    ph( x)    B( x)    V( x)

Proof of Theorem elab2g
StepHypRef Expression
1 elab2g.2 . . 3  |-  B  =  { x  |  ph }
21eleq2i 2502 . 2  |-  ( A  e.  B  <->  A  e.  { x  |  ph }
)
3 elab2g.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
43elabg 3085 . 2  |-  ( A  e.  V  ->  ( A  e.  { x  |  ph }  <->  ps )
)
52, 4syl5bb 250 1  |-  ( A  e.  V  ->  ( A  e.  B  <->  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726   {cab 2424
This theorem is referenced by:  elab2  3087  elab4g  3088  eldif  3332  elun  3490  elin  3532  elprg  3833  elsncg  3838  eluni  4020  eliun  4099  eliin  4100  elopab  4464  elong  4591  elrn2g  5063  eldmg  5067  elrnmpt  5119  elrnmpt1  5121  elimag  5209  elrnmpt2g  6184  eloprabi  6415  tfrlem12  6652  elqsg  6958  elixp2  7068  isacn  7927  isfin1a  8174  isfin2  8176  isfin4  8179  isfin7  8183  isfin3ds  8211  elwina  8563  elina  8564  iswun  8581  eltskg  8627  elgrug  8669  elnp  8866  elnpi  8867  iscat  13899  isps  14636  isdir  14679  elsymgbas2  15098  istopg  16970  isbasisg  17014  isufl  17947  isusp  18293  2sqlem9  21159  isplig  21767  isgrpo  21786  isass  21906  isexid  21907  ismgm  21910  elghomlem2  21952  elunop  23377  adjeu  23394  ballotlemfmpn  24754  dfon2lem3  25414  orderseqlem  25529  elno  25603  elaltxp  25822  isptfin  26377  heiborlem1  26522  heiborlem10  26531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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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