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Theorem ssind 3197
Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
ssind.1  |-  ( ph  ->  A  C_  B )
ssind.2  |-  ( ph  ->  A  C_  C )
Assertion
Ref Expression
ssind  |-  ( ph  ->  A  C_  ( B  i^i  C ) )

Proof of Theorem ssind
StepHypRef Expression
1 ssind.1 . 2  |-  ( ph  ->  A  C_  B )
2 ssind.2 . 2  |-  ( ph  ->  A  C_  C )
3 ssin 3195 . . 3  |-  ( ( A  C_  B  /\  A  C_  C )  <->  A  C_  ( B  i^i  C ) )
43biimpi 118 . 2  |-  ( ( A  C_  B  /\  A  C_  C )  ->  A  C_  ( B  i^i  C ) )
51, 2, 4syl2anc 403 1  |-  ( ph  ->  A  C_  ( B  i^i  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    i^i cin 2973    C_ wss 2974
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987
This theorem is referenced by: (None)
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