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Theorem elin3 3448
Description: Membership in a class defined as a ternary intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin3.x  |-  X  =  ( ( B  i^i  C )  i^i  D )
Assertion
Ref Expression
elin3  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D ) )

Proof of Theorem elin3
StepHypRef Expression
1 elin 3446 . . 3  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21anbi1i 676 . 2  |-  ( ( A  e.  ( B  i^i  C )  /\  A  e.  D )  <->  ( ( A  e.  B  /\  A  e.  C
)  /\  A  e.  D ) )
3 elin3.x . . 3  |-  X  =  ( ( B  i^i  C )  i^i  D )
43elin2 3447 . 2  |-  ( A  e.  X  <->  ( A  e.  ( B  i^i  C
)  /\  A  e.  D ) )
5 df-3an 937 . 2  |-  ( ( A  e.  B  /\  A  e.  C  /\  A  e.  D )  <->  ( ( A  e.  B  /\  A  e.  C
)  /\  A  e.  D ) )
62, 4, 53bitr4i 268 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    i^i cin 3237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-in 3245
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