Theorem elrab3 2750

Description: Membership in a restricted class abstraction, using implicit substitution. (Contributed by NM, 5-Oct-2006.)

Hypothesis

Ref	Expression
elrab.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
elrab3	|- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )

Distinct variable groups:   ps, x   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem elrab3

Step	Hyp	Ref	Expression
1		elrab.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	elrab 2749	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3	2	baib 861	1 |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   {crab 2352

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rab 2357  df-v 2603

This theorem is referenced by:  unimax  3635  frind  4107  ordtriexmidlem2  4264  ordtriexmid  4265  ordtri2orexmid  4266  onsucelsucexmid  4273  0elsucexmid  4308  ordpwsucexmid  4313  ordtri2or2exmid  4314  acexmidlema  5523  acexmidlemb  5524  isnumi  6451  genpelvl  6702  genpelvu  6703  cauappcvgprlemladdru  6846  cauappcvgprlem1  6849  caucvgprlem1  6869  supinfneg  8683  infsupneg  8684  supminfex  8685  ublbneg  8698  negm  8700  infssuzex  10345  gcddvds  10355  dvdslegcd  10356  bezoutlemsup  10398  lcmval  10445  dvdslcm  10451  isprm2lem  10498