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Theorem elin3 3360
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 3358 . . 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 3359 . 2  |-  ( A  e.  X  <->  ( A  e.  ( B  i^i  C
)  /\  A  e.  D ) )
5 df-3an 936 . 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 934    = wceq 1623    e. wcel 1684    i^i cin 3151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159
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