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Theorem dfint2 3720
Description: Alternate definition of class intersection. (Contributed by NM, 28-Jun-1998.)
Assertion
Ref Expression
dfint2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
Distinct variable group:    x, y, A

Proof of Theorem dfint2
StepHypRef Expression
1 df-int 3719 . 2  |-  |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
2 df-ral 2380 . . 3  |-  ( A. y  e.  A  x  e.  y  <->  A. y ( y  e.  A  ->  x  e.  y ) )
32abbii 2215 . 2  |-  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
41, 3eqtr4i 2123 1  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1297    = wceq 1299    e. wcel 1448   {cab 2086   A.wral 2375   |^|cint 3718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-ral 2380  df-int 3719
This theorem is referenced by:  inteq  3721  nfint  3728  intiin  3814
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