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Theorem elini 3166
Description: Membership in an intersection of two classes. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elini.1  |-  A  e.  B
elini.2  |-  A  e.  C
Assertion
Ref Expression
elini  |-  A  e.  ( B  i^i  C
)

Proof of Theorem elini
StepHypRef Expression
1 elini.1 . 2  |-  A  e.  B
2 elini.2 . 2  |-  A  e.  C
3 elin 3165 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
41, 2, 3mpbir2an 884 1  |-  A  e.  ( B  i^i  C
)
Colors of variables: wff set class
Syntax hints:    e. wcel 1434    i^i cin 2981
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 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988
This theorem is referenced by: (None)
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