Theorem elab2g 2924

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 2273	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2923	. 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 1373   e. wcel 2177   {cab 2192

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775

This theorem is referenced by:  elab2  2925  elab4g  2926  eldif  3179  elun  3318  elin  3360  elif  3587  elsng  3653  elprg  3658  eluni  3859  eliun  3937  eliin  3938  elopab  4312  elong  4428  opeliunxp  4738  elrn2g  4876  eldmg  4882  elrnmpt  4936  elrnmpt1  4938  elimag  5035  elrnmpog  6071  eloprabi  6295  tfrlem3ag  6408  tfr1onlem3ag  6436  tfrcllemsucaccv  6453  elqsg  6685  elixp2  6802  isomni  7253  ismkv  7270  iswomni  7282  isacnm  7331  1idprl  7723  1idpru  7724  recexprlemell  7755  recexprlemelu  7756  mertenslemub  11920  mertenslemi1  11921  mertenslem2  11922  4sqexercise1  12796  4sqexercise2  12797  4sqlemsdc  12798  ismgm  13264  istopg  14546  isbasisg  14591  2sqlem8  15675  2sqlem9  15676  isuhgrm  15742  isushgrm  15743  isupgren  15766  isumgren  15776