Theorem elab2g 2868

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 2231	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2867	. 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 1342   e. wcel 2135   {cab 2150

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723

This theorem is referenced by:  elab2  2869  elab4g  2870  eldif  3120  elun  3258  elin  3300  elsng  3585  elprg  3590  eluni  3786  eliun  3864  eliin  3865  elopab  4230  elong  4345  opeliunxp  4653  elrn2g  4788  eldmg  4793  elrnmpt  4847  elrnmpt1  4849  elimag  4944  elrnmpog  5945  eloprabi  6156  tfrlem3ag  6268  tfr1onlem3ag  6296  tfrcllemsucaccv  6313  elqsg  6542  elixp2  6659  isomni  7091  ismkv  7108  iswomni  7120  1idprl  7522  1idpru  7523  recexprlemell  7554  recexprlemelu  7555  mertenslemub  11461  mertenslemi1  11462  mertenslem2  11463  istopg  12538  isbasisg  12583