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Theorem bnj1153 28898
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1153.1  |-  ( ph  ->  X  e.  A )
bnj1153.2  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
bnj1153  |-  ( ph  ->  X  e.  ( A  i^i  B ) )

Proof of Theorem bnj1153
StepHypRef Expression
1 bnj1153.1 . 2  |-  ( ph  ->  X  e.  A )
2 bnj1153.2 . 2  |-  ( ph  ->  X  e.  B )
3 elin 3360 . 2  |-  ( X  e.  ( A  i^i  B )  <->  ( X  e.  A  /\  X  e.  B ) )
41, 2, 3sylanbrc 645 1  |-  ( ph  ->  X  e.  ( A  i^i  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1686    i^i cin 3153
This theorem is referenced by:  bnj1379  28936  bnj1177  29109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-in 3161
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