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Theorem rabssdv 3383
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 2748 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  x  e.  B ) )
4 rabss 3380 . 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 1721   A.wral 2666   {crab 2670    C_ wss 3280
This theorem is referenced by:  suppss2  6259  oemapvali  7596  cantnflem1  7601  harval2  7840  zsupss  10521  ramub1lem1  13349  efgsfo  15326  ablfacrp  15579  ablfac1eu  15586  pgpfac1lem5  15592  ablfaclem3  15600  nrmr0reg  17734  ptcmplem3  18038  abelthlem2  20301  lgamgulmlem1  24766  neibastop2lem  26279  topmeet  26283  cntotbnd  26395  symggen  27279  mapdrvallem2  32128
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rab 2675  df-in 3287  df-ss 3294
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