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Theorem dfin5 3048
Description: Alternate definition for the intersection of two classes. (Contributed by NM, 6-Jul-2005.)
Assertion
Ref Expression
dfin5  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
Distinct variable groups:    x, A    x, B

Proof of Theorem dfin5
StepHypRef Expression
1 df-in 3047 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
2 df-rab 2402 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
31, 2eqtr4i 2141 1  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    e. wcel 1465   {cab 2103   {crab 2397    i^i cin 3040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-4 1472  ax-17 1491  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-rab 2402  df-in 3047
This theorem is referenced by:  nfin  3252  rabbi2dva  3254  ssfidc  6791  znnen  11838  bj-inex  13032
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