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 ) )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224

This proof depends on 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 proof 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 used by:  elab2  2974  elab4g  2975  eldif  3229  elun  3370  elin  3412  elif  3652  elsng  3724  elprg  3729  eluni  3938  eliun  4016  eliin  4017  elopab  4402  elong  4520  opeliunxp  4832  elrn2g  4972  eldmg  4978  elrnmpt  5033  elrnmpt1  5035  elimag  5133  elrnmpog  6204  eloprabi  6436  tfrlem3ag  6585  tfr1onlem3ag  6613  tfrcllemsucaccv  6630  elqsg  6864  elixp2  6990  isomni  7482  ismkv  7499  iswomni  7511  isacnm  7565  1idprl  7963  1idpru  7964  recexprlemell  7995  recexprlemelu  7996  mertenslemub  12325  mertenslemi1  12326  mertenslem2  12327  4sqexercise1  13205  4sqexercise2  13206  4sqlemsdc  13207  ballotfilemfmpn  13291  ismgm  13735  istopg  15242  isbasisg  15287  2sqlem8  16464  2sqlem9  16465  isuhgrm  16534  isushgrm  16535  isupgren  16558  isumgren  16568  isuspgren  16620  isusgren  16621