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Theorem dfint2 3880
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 3879 . 2  |-  |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
2 df-ral 2561 . . 3  |-  ( A. y  e.  A  x  e.  y  <->  A. y ( y  e.  A  ->  x  e.  y ) )
32abbii 2408 . 2  |-  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
41, 3eqtr4i 2319 1  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   |^|cint 3878
This theorem is referenced by:  inteq  3881  nfint  3888  intiin  3972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-ral 2561  df-int 3879
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