Theorem elab2g 2953

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 2298	. 2 |- ( A e. B <-> A e. { x | ph } )
3		elab2g.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
4	3	elabg 2952	. 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 1397   e. wcel 2202   {cab 2217

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  elab2  2954  elab4g  2955  eldif  3209  elun  3348  elin  3390  elif  3617  elsng  3684  elprg  3689  eluni  3896  eliun  3974  eliin  3975  elopab  4352  elong  4470  opeliunxp  4781  elrn2g  4920  eldmg  4926  elrnmpt  4981  elrnmpt1  4983  elimag  5080  elrnmpog  6134  eloprabi  6361  tfrlem3ag  6475  tfr1onlem3ag  6503  tfrcllemsucaccv  6520  elqsg  6754  elixp2  6871  isomni  7335  ismkv  7352  iswomni  7364  isacnm  7418  1idprl  7810  1idpru  7811  recexprlemell  7842  recexprlemelu  7843  mertenslemub  12113  mertenslemi1  12114  mertenslem2  12115  4sqexercise1  12989  4sqexercise2  12990  4sqlemsdc  12991  ismgm  13458  istopg  14742  isbasisg  14787  2sqlem8  15871  2sqlem9  15872  isuhgrm  15941  isushgrm  15942  isupgren  15965  isumgren  15975  isuspgren  16027  isusgren  16028