Theorem elab2g 2884

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 2244	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2883	. 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 1353   e. wcel 2148   {cab 2163

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739

This theorem is referenced by:  elab2  2885  elab4g  2886  eldif  3138  elun  3276  elin  3318  elsng  3606  elprg  3611  eluni  3810  eliun  3888  eliin  3889  elopab  4254  elong  4369  opeliunxp  4677  elrn2g  4812  eldmg  4817  elrnmpt  4871  elrnmpt1  4873  elimag  4969  elrnmpog  5980  eloprabi  6190  tfrlem3ag  6303  tfr1onlem3ag  6331  tfrcllemsucaccv  6348  elqsg  6578  elixp2  6695  isomni  7127  ismkv  7144  iswomni  7156  1idprl  7567  1idpru  7568  recexprlemell  7599  recexprlemelu  7600  mertenslemub  11513  mertenslemi1  11514  mertenslem2  11515  ismgm  12655  istopg  13130  isbasisg  13175  2sqlem8  14092  2sqlem9  14093