Theorem elab2g 2689

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 2104	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2688	. 2 |- ( A e. V -> ( A e. { x | ph } <-> ps ) )
5	2, 4	syl5bb 181	1 |- ( A e. V -> ( A e. B <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   {cab 2026

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559

This theorem is referenced by:  elab2  2690  elab4g  2691  eldif  2927  elun  3084  elin  3126  elsng  3390  elprg  3395  eluni  3583  eliun  3661  eliin  3662  elopab  3995  elong  4110  opeliunxp  4395  elrn2g  4525  eldmg  4530  elrnmpt  4583  elrnmpt1  4585  elimag  4672  elrnmpt2g  5613  eloprabi  5822  tfrlem3ag  5924  elqsg  6156  1idprl  6688  1idpru  6689  recexprlemell  6720  recexprlemelu  6721