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Theorem ss2rabdv 3254
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 2626 . 2  |-  ( ph  ->  A. x  e.  A  ( ps  ->  ch )
)
3 ss2rab 3249 . 2  |-  ( { x  e.  A  |  ps }  C_  { x  e.  A  |  ch } 
<-> 
A. x  e.  A  ( ps  ->  ch )
)
42, 3sylibr 203 1  |-  ( ph  ->  { x  e.  A  |  ps }  C_  { x  e.  A  |  ch } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   A.wral 2543   {crab 2547    C_ wss 3152
This theorem is referenced by:  sess1  4361  reusv6OLD  4545  suppssfv  6074  suppssov1  6075  harword  7279  mrcss  13518  ablfac1b  15305  lspss  15741  aspss  16072  clsss  16791  divstgpopn  17802  metss2lem  18057  equivcau  18726  ovolsslem  18843  itg2monolem1  19105  sqff1o  20420  musum  20431  suppss2f  23201  orvclteinc  23676  lfinpfin  26303  dsmmacl  27207  dsmmsubg  27209  dsmmlss  27210  pclssN  30083  2polssN  30104  dihglblem3N  31485  dochss  31555  mapdordlem2  31827
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-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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