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Theorem dfin5 3004
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 3003 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
2 df-rab 2368 . 2  |-  { x  e.  A  |  x  e.  B }  =  {
x  |  ( x  e.  A  /\  x  e.  B ) }
31, 2eqtr4i 2111 1  |-  ( A  i^i  B )  =  { x  e.  A  |  x  e.  B }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   {cab 2074   {crab 2363    i^i cin 2996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-4 1445  ax-17 1464  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-cleq 2081  df-rab 2368  df-in 3003
This theorem is referenced by:  nfin  3204  rabbi2dva  3206  ssfidc  6623  znnen  11293  bj-inex  11444
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