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Theorem rabssdv 3423
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 1152 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  x  e.  B ) ) )
32ralrimiv 2788 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  x  e.  B ) )
4 rabss 3420 . 2  |-  ( { x  e.  A  |  ps }  C_  B  <->  A. x  e.  A  ( ps  ->  x  e.  B ) )
53, 4sylibr 204 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    e. wcel 1725   A.wral 2705   {crab 2709    C_ wss 3320
This theorem is referenced by:  suppss2  6300  oemapvali  7640  cantnflem1  7645  harval2  7884  zsupss  10565  ramub1lem1  13394  efgsfo  15371  ablfacrp  15624  ablfac1eu  15631  pgpfac1lem5  15637  ablfaclem3  15645  nrmr0reg  17781  ptcmplem3  18085  abelthlem2  20348  lgamgulmlem1  24813  neibastop2lem  26389  topmeet  26393  cntotbnd  26505  symggen  27388  mapdrvallem2  32443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rab 2714  df-in 3327  df-ss 3334
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