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Theorem rabssdv 3429
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 1181 . . 3  |-  ( ph  ->  ( x  e.  A  ->  ( ps  ->  x  e.  B ) ) )
32ralrimiv 2796 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  x  e.  B ) )
4 rabss 3426 . 2  |-  ( { x  e.  A  |  ps }  C_  B  <->  A. x  e.  A  ( ps  ->  x  e.  B ) )
53, 4sylibr 212 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 960    e. wcel 1761   A.wral 2713   {crab 2717    C_ wss 3325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rab 2722  df-in 3332  df-ss 3339
This theorem is referenced by:  suppss2OLD  6314  suppss2  6722  oemapvali  7888  cantnflem1  7893  cantnflem1OLD  7916  harval2  8163  zsupss  10940  ramub1lem1  14083  symggen  15969  efgsfo  16229  ablfacrp  16557  ablfac1eu  16564  pgpfac1lem5  16570  ablfaclem3  16578  nrmr0reg  19281  ptcmplem3  19585  abelthlem2  21856  lgamgulmlem1  26945  neibastop2lem  28506  topmeet  28510  cntotbnd  28620  mapdrvallem2  35012
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