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Theorem ssrab 3251
Description: Subclass of a restricted class abstraction. (Contributed by NM, 16-Aug-2006.)
Assertion
Ref Expression
ssrab  |-  ( B 
C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    ph( x)

Proof of Theorem ssrab
StepHypRef Expression
1 df-rab 2552 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq2i 3203 . 2  |-  ( B 
C_  { x  e.  A  |  ph }  <->  B 
C_  { x  |  ( x  e.  A  /\  ph ) } )
3 ssab 3243 . 2  |-  ( B 
C_  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( x  e.  A  /\  ph ) ) )
4 dfss3 3170 . . . 4  |-  ( B 
C_  A  <->  A. x  e.  B  x  e.  A )
54anbi1i 676 . . 3  |-  ( ( B  C_  A  /\  A. x  e.  B  ph ) 
<->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
6 r19.26 2675 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
7 df-ral 2548 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  A. x ( x  e.  B  ->  (
x  e.  A  /\  ph ) ) )
85, 6, 73bitr2ri 265 . 2  |-  ( A. x ( x  e.  B  ->  ( x  e.  A  /\  ph )
)  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
92, 3, 83bitri 262 1  |-  ( B 
C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547    C_ wss 3152
This theorem is referenced by:  ssrabdv  3252  omssnlim  4670  ordtypelem2  7234  ordtypelem10  7242  card2inf  7269  r0weon  7640  ramtlecl  13047  sscntz  14802  ppttop  16744  epttop  16746  cmpcov2  17117  tgcmp  17128  xkoinjcn  17381  fbssfi  17532  filssufilg  17606  uffixfr  17618  tmdgsum2  17779  symgtgp  17784  ghmcnp  17797  blcls  18052  clsocv  18677  lhop1lem  19360  ressatans  20230  imambfm  23567  conpcon  23766  cvmlift2lem11  23844  cvmlift2lem12  23845  axcontlem3  24594  axcontlem4  24595  hbtlem6  27333
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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