Theorem elab2g 2950

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 2296	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2949	. 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 1395   e. wcel 2200   {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  elab2  2951  elab4g  2952  eldif  3206  elun  3345  elin  3387  elif  3614  elsng  3681  elprg  3686  eluni  3890  eliun  3968  eliin  3969  elopab  4345  elong  4463  opeliunxp  4773  elrn2g  4911  eldmg  4917  elrnmpt  4972  elrnmpt1  4974  elimag  5071  elrnmpog  6116  eloprabi  6340  tfrlem3ag  6453  tfr1onlem3ag  6481  tfrcllemsucaccv  6498  elqsg  6730  elixp2  6847  isomni  7299  ismkv  7316  iswomni  7328  isacnm  7381  1idprl  7773  1idpru  7774  recexprlemell  7805  recexprlemelu  7806  mertenslemub  12040  mertenslemi1  12041  mertenslem2  12042  4sqexercise1  12916  4sqexercise2  12917  4sqlemsdc  12918  ismgm  13385  istopg  14667  isbasisg  14712  2sqlem8  15796  2sqlem9  15797  isuhgrm  15865  isushgrm  15866  isupgren  15889  isumgren  15899  isuspgren  15949  isusgren  15950