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Theorem bnj1153 29141
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 3371 . 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 1696    i^i cin 3164
This theorem is referenced by:  bnj1379  29179  bnj1177  29352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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