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  3891  eliun  3969  eliin  3970  elopab  4346  elong  4464  opeliunxp  4774  elrn2g  4912  eldmg  4918  elrnmpt  4973  elrnmpt1  4975  elimag  5072  elrnmpog  6117  eloprabi  6342  tfrlem3ag  6455  tfr1onlem3ag  6483  tfrcllemsucaccv  6500  elqsg  6732  elixp2  6849  isomni  7303  ismkv  7320  iswomni  7332  isacnm  7385  1idprl  7777  1idpru  7778  recexprlemell  7809  recexprlemelu  7810  mertenslemub  12045  mertenslemi1  12046  mertenslem2  12047  4sqexercise1  12921  4sqexercise2  12922  4sqlemsdc  12923  ismgm  13390  istopg  14673  isbasisg  14718  2sqlem8  15802  2sqlem9  15803  isuhgrm  15871  isushgrm  15872  isupgren  15895  isumgren  15905  isuspgren  15955  isusgren  15956