Theorem elrab3 2930

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 2929	. 2 |- ( A e. { x e. B | ph } <-> ( A e. B /\ ps ) )
3	2	baib 921	1 |- ( A e. B -> ( A e. { x e. B | ph } <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2176   {crab 2488

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rab 2493  df-v 2774

This theorem is referenced by:  unimax  3884  undifexmid  4237  frind  4399  ordtriexmidlem2  4568  ordtriexmid  4569  ontriexmidim  4570  ordtri2orexmid  4571  onsucelsucexmid  4578  0elsucexmid  4613  ordpwsucexmid  4618  ordtri2or2exmid  4619  ontri2orexmidim  4620  canth  5897  acexmidlema  5935  acexmidlemb  5936  isnumi  7289  genpelvl  7625  genpelvu  7626  cauappcvgprlemladdru  7769  cauappcvgprlem1  7772  caucvgprlem1  7792  sup3exmid  9030  supinfneg  9716  infsupneg  9717  supminfex  9718  ublbneg  9734  negm  9736  infssuzex  10376  hashinfuni  10922  gcddvds  12284  dvdslegcd  12285  bezoutlemsup  12330  uzwodc  12358  lcmval  12385  dvdslcm  12391  isprm2lem  12438