Theorem elab2g 2683

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 2101	. 2 |- (A e. B <-> A e. {x | ph })
3		elab2g.1	. . 3 |- (x = A -> (ph <-> ps))
4	3	elabg 2682	. 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 1242   e. wcel 1390   {cab 2023

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553

This theorem is referenced by:  elab2  2684  elab4g  2685  eldif  2921  elun  3078  elin  3120  elprg  3384  elsncg  3389  eluni  3574  eliun  3652  eliin  3653  elopab  3986  elong  4076  opeliunxp  4338  elrn2g  4468  eldmg  4473  elrnmpt  4526  elrnmpt1  4528  elimag  4615  elrnmpt2g  5555  eloprabi  5764  tfrlem3ag  5865  elqsg  6092  1idprl  6566  1idpru  6567  recexprlemell  6594  recexprlemelu  6595