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  6584  tfr1onlem3ag  6612  tfrcllemsucaccv  6629  elqsg  6863  elixp2  6988  isomni  7482  ismkv  7499  iswomni  7511  isacnm  7565  1idprl  7963  1idpru  7964  recexprlemell  7995  recexprlemelu  7996  mertenslemub  12326  mertenslemi1  12327  mertenslem2  12328  4sqexercise1  13206  4sqexercise2  13207  4sqlemsdc  13208  ballotfilemfmpn  13292  ismgm  13736  istopg  15225  isbasisg  15270  2sqlem8  16447  2sqlem9  16448  isuhgrm  16517  isushgrm  16518  isupgren  16541  isumgren  16551  isuspgren  16603  isusgren  16604