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Theorem elrint2 3941
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint2  |-  ( X  e.  A  ->  ( X  e.  ( A  i^i  |^| B )  <->  A. y  e.  B  X  e.  y ) )
Distinct variable groups:    y, B    y, X
Allowed substitution hint:    A( y)

Proof of Theorem elrint2
StepHypRef Expression
1 elrint 3940 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
21baib 871 1  |-  ( X  e.  A  ->  ( X  e.  ( A  i^i  |^| B )  <->  A. y  e.  B  X  e.  y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    e. wcel 1701   A.wral 2577    i^i cin 3185   |^|cint 3899
This theorem is referenced by:  mreacs  13609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ral 2582  df-v 2824  df-in 3193  df-int 3900
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