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  4400  elong  4518  opeliunxp  4830  elrn2g  4970  eldmg  4976  elrnmpt  5031  elrnmpt1  5033  elimag  5130  elrnmpog  6201  eloprabi  6432  tfrlem3ag  6580  tfr1onlem3ag  6608  tfrcllemsucaccv  6625  elqsg  6859  elixp2  6984  isomni  7477  ismkv  7494  iswomni  7506  isacnm  7560  1idprl  7958  1idpru  7959  recexprlemell  7990  recexprlemelu  7991  mertenslemub  12320  mertenslemi1  12321  mertenslem2  12322  4sqexercise1  13200  4sqexercise2  13201  4sqlemsdc  13202  ballotfilemfmpn  13286  ismgm  13730  istopg  15191  isbasisg  15236  2sqlem8  16408  2sqlem9  16409  isuhgrm  16478  isushgrm  16479  isupgren  16502  isumgren  16512  isuspgren  16564  isusgren  16565