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Theorem rabssdv 3253
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 2625 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  x  e.  B ) )
4 rabss 3250 . 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 1684   A.wral 2543   {crab 2547    C_ wss 3152
This theorem is referenced by:  suppss2  6073  oemapvali  7386  cantnflem1  7391  harval2  7630  zsupss  10307  ramub1lem1  13073  efgsfo  15048  ablfacrp  15301  ablfac1eu  15308  pgpfac1lem5  15314  ablfaclem3  15322  nrmr0reg  17440  ptcmplem3  17748  abelthlem2  19808  rayline  26156  neibastop2lem  26309  topmeet  26313  cntotbnd  26520  symggen  27411  mapdrvallem2  31835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rab 2552  df-in 3159  df-ss 3166
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