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Theorem dfint2 4228
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 4227 . 2  |-  |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
2 df-ral 2761 . . 3  |-  ( A. y  e.  A  x  e.  y  <->  A. y ( y  e.  A  ->  x  e.  y ) )
32abbii 2587 . 2  |-  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
41, 3eqtr4i 2496 1  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   |^|cint 4226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-ral 2761  df-int 4227
This theorem is referenced by:  inteq  4229  nfint  4236  intss  4247  intiin  4323  dfint3  30790
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