Theorem elab2g 2967

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 2301	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2966	. 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 2205   {cab 2220

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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817

This theorem is referenced by:  elab2  2968  elab4g  2969  eldif  3223  elun  3364  elin  3406  elif  3638  elsng  3709  elprg  3714  eluni  3922  eliun  4000  eliin  4001  elopab  4381  elong  4499  opeliunxp  4810  elrn2g  4950  eldmg  4956  elrnmpt  5011  elrnmpt1  5013  elimag  5110  elrnmpog  6174  eloprabi  6405  tfrlem3ag  6553  tfr1onlem3ag  6581  tfrcllemsucaccv  6598  elqsg  6832  elixp2  6950  isomni  7440  ismkv  7457  iswomni  7469  isacnm  7523  1idprl  7921  1idpru  7922  recexprlemell  7953  recexprlemelu  7954  mertenslemub  12245  mertenslemi1  12246  mertenslem2  12247  4sqexercise1  13121  4sqexercise2  13122  4sqlemsdc  13123  ballotfilemfmpn  13178  ismgm  13620  istopg  14990  isbasisg  15035  2sqlem8  16122  2sqlem9  16123  isuhgrm  16192  isushgrm  16193  isupgren  16216  isumgren  16226  isuspgren  16278  isusgren  16279