Theorem elab2g 2964

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

Step	Hyp	Ref	Expression
1		elab2g.2	. . 3 |- B = { x | ph }
2	1	eleq2i 2299	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2963	. 2 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
5	2, 4	bitrid 192	1 |- ( A e. V -> ( A e. B <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2203   {cab 2218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815

This theorem is referenced by:  elab2  2965  elab4g  2966  eldif  3220  elun  3360  elin  3402  elif  3634  elsng  3704  elprg  3709  eluni  3917  eliun  3995  eliin  3996  elopab  4376  elong  4494  opeliunxp  4805  elrn2g  4945  eldmg  4951  elrnmpt  5006  elrnmpt1  5008  elimag  5105  elrnmpog  6166  eloprabi  6392  tfrlem3ag  6540  tfr1onlem3ag  6568  tfrcllemsucaccv  6585  elqsg  6819  elixp2  6937  isomni  7427  ismkv  7444  iswomni  7456  isacnm  7510  1idprl  7905  1idpru  7906  recexprlemell  7937  recexprlemelu  7938  mertenslemub  12220  mertenslemi1  12221  mertenslem2  12222  4sqexercise1  13096  4sqexercise2  13097  4sqlemsdc  13098  ismgm  13570  istopg  14864  isbasisg  14909  2sqlem8  15996  2sqlem9  15997  isuhgrm  16066  isushgrm  16067  isupgren  16090  isumgren  16100  isuspgren  16152  isusgren  16153