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Theorem ssrab 3578
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 2823 . . 3  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21sseq2i 3529 . 2  |-  ( B 
C_  { x  e.  A  |  ph }  <->  B 
C_  { x  |  ( x  e.  A  /\  ph ) } )
3 ssab 3570 . 2  |-  ( B 
C_  { x  |  ( x  e.  A  /\  ph ) }  <->  A. x
( x  e.  B  ->  ( x  e.  A  /\  ph ) ) )
4 dfss3 3494 . . . 4  |-  ( B 
C_  A  <->  A. x  e.  B  x  e.  A )
54anbi1i 695 . . 3  |-  ( ( B  C_  A  /\  A. x  e.  B  ph ) 
<->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
6 r19.26 2989 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  ( A. x  e.  B  x  e.  A  /\  A. x  e.  B  ph ) )
7 df-ral 2819 . . 3  |-  ( A. x  e.  B  (
x  e.  A  /\  ph )  <->  A. x ( x  e.  B  ->  (
x  e.  A  /\  ph ) ) )
85, 6, 73bitr2ri 274 . 2  |-  ( A. x ( x  e.  B  ->  ( x  e.  A  /\  ph )
)  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
92, 3, 83bitri 271 1  |-  ( B 
C_  { x  e.  A  |  ph }  <->  ( B  C_  A  /\  A. x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1377    e. wcel 1767   {cab 2452   A.wral 2814   {crab 2818    C_ wss 3476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rab 2823  df-in 3483  df-ss 3490
This theorem is referenced by:  ssrabdv  3579  omssnlim  6699  ordtypelem2  7945  ordtypelem10  7953  card2inf  7982  r0weon  8391  ramtlecl  14380  sscntz  16178  ppttop  19314  epttop  19316  cmpcov2  19696  tgcmp  19707  xkoinjcn  20015  fbssfi  20165  filssufilg  20239  uffixfr  20251  tmdgsum2  20422  symgtgp  20427  ghmcnp  20440  blcls  20836  clsocv  21517  lhop1lem  22241  ressatans  23090  axcontlem3  24042  axcontlem4  24043  imambfm  27984  conpcon  28431  cvmlift2lem11  28509  cvmlift2lem12  28510  hbtlem6  30909  bj-rabtr  33795
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