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Theorem dfint2 4257
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 4256 . 2  |-  |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
2 df-ral 2776 . . 3  |-  ( A. y  e.  A  x  e.  y  <->  A. y ( y  e.  A  ->  x  e.  y ) )
32abbii 2551 . 2  |-  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
41, 3eqtr4i 2454 1  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1435    = wceq 1437    e. wcel 1872   {cab 2407   A.wral 2771   |^|cint 4255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-ral 2776  df-int 4256
This theorem is referenced by:  inteq  4258  nfint  4265  intss  4276  intiin  4353  dfint3  30724
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