Theorem elab2g 2873

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 2233	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2872	. 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 1343   e. wcel 2136   {cab 2151

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  elab2  2874  elab4g  2875  eldif  3125  elun  3263  elin  3305  elsng  3591  elprg  3596  eluni  3792  eliun  3870  eliin  3871  elopab  4236  elong  4351  opeliunxp  4659  elrn2g  4794  eldmg  4799  elrnmpt  4853  elrnmpt1  4855  elimag  4950  elrnmpog  5954  eloprabi  6164  tfrlem3ag  6277  tfr1onlem3ag  6305  tfrcllemsucaccv  6322  elqsg  6551  elixp2  6668  isomni  7100  ismkv  7117  iswomni  7129  1idprl  7531  1idpru  7532  recexprlemell  7563  recexprlemelu  7564  mertenslemub  11475  mertenslemi1  11476  mertenslem2  11477  ismgm  12588  istopg  12647  isbasisg  12692  2sqlem8  13609  2sqlem9  13610