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  3607  elprg  3612  eluni  3812  eliun  3890  eliin  3891  elopab  4258  elong  4373  opeliunxp  4681  elrn2g  4817  eldmg  4822  elrnmpt  4876  elrnmpt1  4878  elimag  4974  elrnmpog  5986  eloprabi  6196  tfrlem3ag  6309  tfr1onlem3ag  6337  tfrcllemsucaccv  6354  elqsg  6584  elixp2  6701  isomni  7133  ismkv  7150  iswomni  7162  1idprl  7588  1idpru  7589  recexprlemell  7620  recexprlemelu  7621  mertenslemub  11541  mertenslemi1  11542  mertenslem2  11543  ismgm  12775  istopg  13469  isbasisg  13514  2sqlem8  14440  2sqlem9  14441