Theorem elab2g 2954

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 2298	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2953	. 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 2202   {cab 2217

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  elab2  2955  elab4g  2956  eldif  3210  elun  3350  elin  3392  elif  3621  elsng  3688  elprg  3693  eluni  3901  eliun  3979  eliin  3980  elopab  4358  elong  4476  opeliunxp  4787  elrn2g  4926  eldmg  4932  elrnmpt  4987  elrnmpt1  4989  elimag  5086  elrnmpog  6144  eloprabi  6370  tfrlem3ag  6518  tfr1onlem3ag  6546  tfrcllemsucaccv  6563  elqsg  6797  elixp2  6914  isomni  7378  ismkv  7395  iswomni  7407  isacnm  7461  1idprl  7853  1idpru  7854  recexprlemell  7885  recexprlemelu  7886  mertenslemub  12158  mertenslemi1  12159  mertenslem2  12160  4sqexercise1  13034  4sqexercise2  13035  4sqlemsdc  13036  ismgm  13503  istopg  14793  isbasisg  14838  2sqlem8  15925  2sqlem9  15926  isuhgrm  15995  isushgrm  15996  isupgren  16019  isumgren  16029  isuspgren  16081  isusgren  16082