MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssrabdv Structured version   Unicode version

Theorem ssrabdv 3429
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 31-Aug-2006.)
Hypotheses
Ref Expression
ssrabdv.1  |-  ( ph  ->  B  C_  A )
ssrabdv.2  |-  ( (
ph  /\  x  e.  B )  ->  ps )
Assertion
Ref Expression
ssrabdv  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem ssrabdv
StepHypRef Expression
1 ssrabdv.1 . 2  |-  ( ph  ->  B  C_  A )
2 ssrabdv.2 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ps )
32ralrimiva 2797 . 2  |-  ( ph  ->  A. x  e.  B  ps )
4 ssrab 3428 . 2  |-  ( B 
C_  { x  e.  A  |  ps }  <->  ( B  C_  A  /\  A. x  e.  B  ps ) )
51, 3, 4sylanbrc 664 1  |-  ( ph  ->  B  C_  { x  e.  A  |  ps } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   A.wral 2713   {crab 2717    C_ wss 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-in 3333  df-ss 3340
This theorem is referenced by:  mrcmndind  15492  symggen  15974  ablfac1eu  16572  lspsolvlem  17221  prdsxmslem2  20102  ovolicc2lem4  21001  abelth2  21905  perfectlem2  22567  cvmlift2lem11  27200  idomsubgmo  29560  bj-rabtrAUTO  32431  mapdrvallem3  35288
  Copyright terms: Public domain W3C validator