Theorem elrab3 2937

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 2936	. 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 2178   {crab 2490

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rab 2495  df-v 2778

This theorem is referenced by:  unimax  3898  undifexmid  4253  frind  4417  ordtriexmidlem2  4586  ordtriexmid  4587  ontriexmidim  4588  ordtri2orexmid  4589  onsucelsucexmid  4596  0elsucexmid  4631  ordpwsucexmid  4636  ordtri2or2exmid  4637  ontri2orexmidim  4638  canth  5920  acexmidlema  5958  acexmidlemb  5959  isnumi  7315  genpelvl  7660  genpelvu  7661  cauappcvgprlemladdru  7804  cauappcvgprlem1  7807  caucvgprlem1  7827  sup3exmid  9065  supinfneg  9751  infsupneg  9752  supminfex  9753  ublbneg  9769  negm  9771  infssuzex  10413  hashinfuni  10959  gcddvds  12399  dvdslegcd  12400  bezoutlemsup  12445  uzwodc  12473  lcmval  12500  dvdslcm  12506  isprm2lem  12553