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Theorem ssin 3268
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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elin 3229 . . . . 5  |-  ( x  e.  ( B  i^i  C )  <->  ( x  e.  B  /\  x  e.  C ) )
21imbi2i 225 . . . 4  |-  ( ( x  e.  A  ->  x  e.  ( B  i^i  C ) )  <->  ( x  e.  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
32albii 1431 . . 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 577 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  ->  x  e.  B )  /\  ( x  e.  A  ->  x  e.  C ) ) )
54albii 1431 . . 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 1442 . . 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 206 . 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 3056 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
9 dfss2 3056 . . 3  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
108, 9anbi12i 455 . 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 3056 . 2  |-  ( A 
C_  ( B  i^i  C )  <->  A. x ( x  e.  A  ->  x  e.  ( B  i^i  C
) ) )
127, 10, 113bitr4i 211 1  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    e. wcel 1465    i^i cin 3040    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054
This theorem is referenced by:  ssini  3269  ssind  3270  uneqin  3297  trin  4006  pwin  4174  peano5  4482  fin  5279  tgval  12145  eltg3i  12152  innei  12259  cnptoprest2  12336
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