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Theorem ssin 3302
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 3263 . . . . 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 1447 . . 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 593 . . . 4  |-  ( ( x  e.  A  -> 
( x  e.  B  /\  x  e.  C
) )  <->  ( (
x  e.  A  ->  x  e.  B )  /\  ( x  e.  A  ->  x  e.  C ) ) )
54albii 1447 . . 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 1458 . . 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 3090 . . 3  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
9 dfss2 3090 . . 3  |-  ( A 
C_  C  <->  A. x
( x  e.  A  ->  x  e.  C ) )
108, 9anbi12i 456 . 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 3090 . 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 1330    e. wcel 1481    i^i cin 3074    C_ wss 3075
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-in 3081  df-ss 3088
This theorem is referenced by:  ssini  3303  ssind  3304  uneqin  3331  trin  4043  pwin  4211  peano5  4519  fin  5316  tgval  12255  eltg3i  12262  innei  12369  cnptoprest2  12446
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