Theorem elab2g 2951

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 2296	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2950	. 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 1395   e. wcel 2200   {cab 2215

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802

This theorem is referenced by:  elab2  2952  elab4g  2953  eldif  3207  elun  3346  elin  3388  elif  3615  elsng  3682  elprg  3687  eluni  3894  eliun  3972  eliin  3973  elopab  4350  elong  4468  opeliunxp  4779  elrn2g  4918  eldmg  4924  elrnmpt  4979  elrnmpt1  4981  elimag  5078  elrnmpog  6129  eloprabi  6356  tfrlem3ag  6470  tfr1onlem3ag  6498  tfrcllemsucaccv  6515  elqsg  6749  elixp2  6866  isomni  7326  ismkv  7343  iswomni  7355  isacnm  7408  1idprl  7800  1idpru  7801  recexprlemell  7832  recexprlemelu  7833  mertenslemub  12085  mertenslemi1  12086  mertenslem2  12087  4sqexercise1  12961  4sqexercise2  12962  4sqlemsdc  12963  ismgm  13430  istopg  14713  isbasisg  14758  2sqlem8  15842  2sqlem9  15843  isuhgrm  15912  isushgrm  15913  isupgren  15936  isumgren  15946  isuspgren  15996  isusgren  15997