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  4238  frind  4400  ordtriexmidlem2  4569  ordtriexmid  4570  ontriexmidim  4571  ordtri2orexmid  4572  onsucelsucexmid  4579  0elsucexmid  4614  ordpwsucexmid  4619  ordtri2or2exmid  4620  ontri2orexmidim  4621  canth  5899  acexmidlema  5937  acexmidlemb  5938  isnumi  7291  genpelvl  7627  genpelvu  7628  cauappcvgprlemladdru  7771  cauappcvgprlem1  7774  caucvgprlem1  7794  sup3exmid  9032  supinfneg  9718  infsupneg  9719  supminfex  9720  ublbneg  9736  negm  9738  infssuzex  10378  hashinfuni  10924  gcddvds  12317  dvdslegcd  12318  bezoutlemsup  12363  uzwodc  12391  lcmval  12418  dvdslcm  12424  isprm2lem  12471