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

Theorem rabssdv 3287
Description: Subclass of a restricted class abstraction (deduction rule). (Contributed by NM, 2-Feb-2015.)
Hypothesis
Ref Expression
rabssdv.1  |-  ( (
ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )
Assertion
Ref Expression
rabssdv  |-  ( ph  ->  { x  e.  A  |  ps }  C_  B
)
Distinct variable groups:    x, B    ph, x
Allowed substitution hints:    ps( x)    A( x)

Proof of Theorem rabssdv
StepHypRef Expression
1 rabssdv.1 . . . 4  |-  ( (
ph  /\  x  e.  A  /\  ps )  ->  x  e.  B )
213exp 1150 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  x  e.  B ) ) )
32ralrimiv 2659 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  x  e.  B ) )
4 rabss 3284 . 2  |-  ( { x  e.  A  |  ps }  C_  B  <->  A. x  e.  A  ( ps  ->  x  e.  B ) )
53, 4sylibr 203 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1701   A.wral 2577   {crab 2581    C_ wss 3186
This theorem is referenced by:  suppss2  6115  oemapvali  7431  cantnflem1  7436  harval2  7675  zsupss  10354  ramub1lem1  13120  efgsfo  15097  ablfacrp  15350  ablfac1eu  15357  pgpfac1lem5  15363  ablfaclem3  15371  nrmr0reg  17496  ptcmplem3  17800  abelthlem2  19861  neibastop2lem  25458  topmeet  25462  cntotbnd  25668  symggen  26559  mapdrvallem2  31653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-rab 2586  df-in 3193  df-ss 3200
  Copyright terms: Public domain W3C validator