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Theorem ss2rabdv 3426
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
ss2rabdv  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch } )
Distinct variable group:    ph, x
Allowed substitution hints:    ps( x)    ch( x)    A( x)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  ( ps  ->  ch ) )
21ralrimiva 2791 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
3 ss2rab 3421 . 2  |-  ( { x  e.  A  |  ps }  C_  { x  e.  A  |  ch } 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
42, 3sylibr 205 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   A.wral 2707   {crab 2711    C_ wss 3322
This theorem is referenced by:  sess1  4553  reusv6OLD  4737  suppssfv  6304  suppssov1  6305  harword  7536  mrcss  13846  ablfac1b  15633  lspss  16065  aspss  16396  clsss  17123  divstgpopn  18154  metss2lem  18546  equivcau  19258  ovolsslem  19385  itg2monolem1  19645  sqff1o  20970  musum  20981  cusgrafilem1  21493  suppss2f  24054  orvclteinc  24738  lgamucov  24827  cnambfre  26266  lfinpfin  26396  dsmmacl  27197  dsmmsubg  27199  dsmmlss  27200  pclssN  30764  2polssN  30785  dihglblem3N  32166  dochss  32236  mapdordlem2  32508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rab 2716  df-in 3329  df-ss 3336
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