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Theorem elin3 3534
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 3532 . . 3  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21anbi1i 678 . 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 3533 . 2  |-  ( A  e.  X  <->  ( A  e.  ( B  i^i  C
)  /\  A  e.  D ) )
5 df-3an 939 . 2  |-  ( ( A  e.  B  /\  A  e.  C  /\  A  e.  D )  <->  ( ( A  e.  B  /\  A  e.  C
)  /\  A  e.  D ) )
62, 4, 53bitr4i 270 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    i^i cin 3321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329
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