Theorem elrab3 2964

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 2963	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3	2	baib 927	1 |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   e. wcel 2202   {crab 2515

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805

This theorem is referenced by:  unimax  3932  undifexmid  4289  frind  4455  ordtriexmidlem2  4624  ordtriexmid  4625  ontriexmidim  4626  ordtri2orexmid  4627  onsucelsucexmid  4634  0elsucexmid  4669  ordpwsucexmid  4674  ordtri2or2exmid  4675  ontri2orexmidim  4676  canth  5979  acexmidlema  6019  acexmidlemb  6020  isnumi  7446  genpelvl  7792  genpelvu  7793  cauappcvgprlemladdru  7936  cauappcvgprlem1  7939  caucvgprlem1  7959  sup3exmid  9196  supinfneg  9890  infsupneg  9891  supminfex  9892  ublbneg  9908  negm  9910  infssuzex  10556  hashinfuni  11102  gcddvds  12614  dvdslegcd  12615  bezoutlemsup  12660  uzwodc  12688  lcmval  12715  dvdslcm  12721  isprm2lem  12768  eupth2lem3lem3fi  16411  eupth2lem3lem6fi  16412  eupth2lem3lem4fi  16414