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Theorem ssin 3392
Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssin  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )

Proof of Theorem ssin
StepHypRef Expression
1 elin 3359 . . . . 5  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21imbi2i 305 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
32albii 1558 . . 3  |-  ( A. x ( x  e.  A  ->  x  e.  ( B  i^i  C ) )  <->  A. x ( x  e.  A  ->  (
x  e.  B  /\  x  e.  C )
) )
4 jcab 836 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  ->  x  e.  B )  /\  ( x  e.  A  ->  x  e.  C ) ) )
54albii 1558 . . 3  |-  ( A. x ( x  e.  A  ->  ( x  e.  B  /\  x  e.  C ) )  <->  A. x
( ( x  e.  A  ->  x  e.  B )  /\  (
x  e.  A  ->  x  e.  C )
) )
6 19.26 1585 . . 3  |-  ( A. x ( ( x  e.  A  ->  x  e.  B )  /\  (
x  e.  A  ->  x  e.  C )
)  <->  ( A. x
( x  e.  A  ->  x  e.  B )  /\  A. x ( x  e.  A  ->  x  e.  C )
) )
73, 5, 63bitrri 265 . 2  |-  ( ( A. x ( x  e.  A  ->  x  e.  B )  /\  A. x ( x  e.  A  ->  x  e.  C ) )  <->  A. x
( x  e.  A  ->  x  e.  ( B  i^i  C ) ) )
8 dfss2 3170 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
9 dfss2 3170 . . 3  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
108, 9anbi12i 681 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  <->  ( A. x ( x  e.  A  ->  x  e.  B )  /\  A. x ( x  e.  A  ->  x  e.  C ) ) )
11 dfss2 3170 . 2  |-  ( A 
C_  ( B  i^i  C )  <->  A. x ( x  e.  A  ->  x  e.  ( B  i^i  C
) ) )
127, 10, 113bitr4i 270 1  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    e. wcel 1688    i^i cin 3152    C_ wss 3153
This theorem is referenced by:  ssini  3393  ssind  3394  uneqin  3421  disjpss  3506  trin  4124  pwin  4296  fin  5386  epfrs  7408  tcmin  7421  resscntz  14801  subgdmdprd  15263  tgval  16687  eltg3i  16693  innei  16856  cnprest2  17012  subislly  17201  lly1stc  17216  xkohaus  17341  xkoinjcn  17375  opnfbas  17531  supfil  17584  rnelfm  17642  tsmsres  17820  chabs2  22088  cmbr4i  22172  pjin3i  22766  mdbr2  22868  dmdbr2  22875  dmdbr5  22880  mdslle1i  22889  mdslle2i  22890  mdslj1i  22891  mdslj2i  22892  mdsl2i  22894  mdslmd1lem1  22897  mdslmd1lem2  22898  mdslmd1i  22901  mdslmd3i  22904  hatomistici  22934  chrelat2i  22937  cvexchlem  22940  mdsymlem1  22975  mdsymlem3  22977  mdsymlem6  22980  dmdbr5ati  22994  ballotlem2  23040  iccllyscon  23185  wfrlem4  23660  frrlem4  23685  dfps2  24688  toplat  24689  clsint  24912  eltintpar  25298  inttaror  25299  pgapspf2  25452  bsstrs  25545  heibor1lem  25932  dochexmidlem1  30917
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-in 3160  df-ss 3167
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