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Theorem elind 3231
Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
elind.1  |-  ( ph  ->  X  e.  A )
elind.2  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
elind  |-  ( ph  ->  X  e.  ( A  i^i  B ) )

Proof of Theorem elind
StepHypRef Expression
1 elind.1 . 2  |-  ( ph  ->  X  e.  A )
2 elind.2 . 2  |-  ( ph  ->  X  e.  B )
3 elin 3229 . 2  |-  ( X  e.  ( A  i^i  B )  <->  ( X  e.  A  /\  X  e.  B ) )
41, 2, 3sylanbrc 413 1  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465    i^i cin 3040
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
This theorem is referenced by:  elfir  6829  infpwfidom  7022  strslfv2d  11912  baspartn  12128  bastg  12141  isopn3  12205  restbasg  12248  lmss  12326  metrest  12586  tgioo  12626  dvmulxxbr  12746  pilem3  12775
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