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Theorem elin3 3492
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 3490 . . 3  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
21anbi1i 677 . 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 3491 . 2  |-  ( A  e.  X  <->  ( A  e.  ( B  i^i  C
)  /\  A  e.  D ) )
5 df-3an 938 . 2  |-  ( ( A  e.  B  /\  A  e.  C  /\  A  e.  D )  <->  ( ( A  e.  B  /\  A  e.  C
)  /\  A  e.  D ) )
62, 4, 53bitr4i 269 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C  /\  A  e.  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    i^i cin 3279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-in 3287
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