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Theorem rabss 3356
Description: Restricted class abstraction in a subclass relationship. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
rabss  |-  ( { x  e.  A  |  ph }  C_  B  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
Distinct variable group:    x, B
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem rabss
StepHypRef Expression
1 df-rab 2651 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq1i 3308 . 2  |-  ( { x  e.  A  |  ph }  C_  B  <->  { x  |  ( x  e.  A  /\  ph ) }  C_  B )
3 abss 3348 . 2  |-  ( { x  |  ( x  e.  A  /\  ph ) }  C_  B  <->  A. x
( ( x  e.  A  /\  ph )  ->  x  e.  B ) )
4 impexp 434 . . . 4  |-  ( ( ( x  e.  A  /\  ph )  ->  x  e.  B )  <->  ( x  e.  A  ->  ( ph  ->  x  e.  B ) ) )
54albii 1572 . . 3  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  e.  B
)  <->  A. x ( x  e.  A  ->  ( ph  ->  x  e.  B
) ) )
6 df-ral 2647 . . 3  |-  ( A. x  e.  A  ( ph  ->  x  e.  B
)  <->  A. x ( x  e.  A  ->  ( ph  ->  x  e.  B
) ) )
75, 6bitr4i 244 . 2  |-  ( A. x ( ( x  e.  A  /\  ph )  ->  x  e.  B
)  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
82, 3, 73bitri 263 1  |-  ( { x  e.  A  |  ph }  C_  B  <->  A. x  e.  A  ( ph  ->  x  e.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    e. wcel 1717   {cab 2366   A.wral 2642   {crab 2646    C_ wss 3256
This theorem is referenced by:  rabssdv  3359  reusv6OLD  4667  fnsuppres  5884  wemapso2  7447  tskwe2  8574  grothac  8631  uzwo3  10494  phibndlem  13079  dfphi2  13083  ramval  13296  gsumvallem1  14691  istopon  16906  ordtrest2lem  17182  filssufilg  17857  cfinufil  17874  blsscls2  18417  nmhmcn  18992  ovolshftlem2  19266  atansssdm  20633  sgmss  20749  sspval  22063  ubthlem2  22214  truae  24381  nnubfi  26138  prnc  26361
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ral 2647  df-rab 2651  df-in 3263  df-ss 3270
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