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Theorem elrint 3655
Description: Membership in a restricted intersection. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
elrint  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
Distinct variable groups:    y, B    y, X
Allowed substitution hint:    A( y)

Proof of Theorem elrint
StepHypRef Expression
1 elin 3126 . 2  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  X  e. 
|^| B ) )
2 elintg 3623 . . 3  |-  ( X  e.  A  ->  ( X  e.  |^| B  <->  A. y  e.  B  X  e.  y ) )
32pm5.32i 427 . 2  |-  ( ( X  e.  A  /\  X  e.  |^| B )  <-> 
( X  e.  A  /\  A. y  e.  B  X  e.  y )
)
41, 3bitri 173 1  |-  ( X  e.  ( A  i^i  |^| B )  <->  ( X  e.  A  /\  A. y  e.  B  X  e.  y ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    e. wcel 1393   A.wral 2306    i^i cin 2916   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-int 3616
This theorem is referenced by:  elrint2  3656
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