Theorem elab2g 2973

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 2305	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2972	. 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 1402   e. wcel 2209   {cab 2224

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823

This theorem is referenced by:  elab2  2974  elab4g  2975  eldif  3229  elun  3370  elin  3412  elif  3649  elsng  3720  elprg  3725  eluni  3933  eliun  4011  eliin  4012  elopab  4395  elong  4513  opeliunxp  4825  elrn2g  4965  eldmg  4971  elrnmpt  5026  elrnmpt1  5028  elimag  5125  elrnmpog  6191  eloprabi  6422  tfrlem3ag  6570  tfr1onlem3ag  6598  tfrcllemsucaccv  6615  elqsg  6849  elixp2  6974  isomni  7466  ismkv  7483  iswomni  7495  isacnm  7549  1idprl  7947  1idpru  7948  recexprlemell  7979  recexprlemelu  7980  mertenslemub  12279  mertenslemi1  12280  mertenslem2  12281  4sqexercise1  13155  4sqexercise2  13156  4sqlemsdc  13157  ballotfilemfmpn  13212  ismgm  13654  istopg  15023  isbasisg  15068  2sqlem8  16156  2sqlem9  16157  isuhgrm  16226  isushgrm  16227  isupgren  16250  isumgren  16260  isuspgren  16312  isusgren  16313