Theorem elab2g 2877

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 2237	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2876	. 2 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
5	2, 4	syl5bb 191	1 |- ( A e. V -> ( A e. B <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   {cab 2156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elab2  2878  elab4g  2879  eldif  3130  elun  3268  elin  3310  elsng  3598  elprg  3603  eluni  3799  eliun  3877  eliin  3878  elopab  4243  elong  4358  opeliunxp  4666  elrn2g  4801  eldmg  4806  elrnmpt  4860  elrnmpt1  4862  elimag  4957  elrnmpog  5965  eloprabi  6175  tfrlem3ag  6288  tfr1onlem3ag  6316  tfrcllemsucaccv  6333  elqsg  6563  elixp2  6680  isomni  7112  ismkv  7129  iswomni  7141  1idprl  7552  1idpru  7553  recexprlemell  7584  recexprlemelu  7585  mertenslemub  11497  mertenslemi1  11498  mertenslem2  11499  ismgm  12611  istopg  12791  isbasisg  12836  2sqlem8  13753  2sqlem9  13754