MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfint2 Structured version   Unicode version

Theorem dfint2 4260
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 4259 . 2  |-  |^| A  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
2 df-ral 2787 . . 3  |-  ( A. y  e.  A  x  e.  y  <->  A. y ( y  e.  A  ->  x  e.  y ) )
32abbii 2563 . 2  |-  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y ( y  e.  A  ->  x  e.  y ) }
41, 3eqtr4i 2461 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 1870   {cab 2414   A.wral 2782   |^|cint 4258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2787  df-int 4259
This theorem is referenced by:  inteq  4261  nfint  4268  intss  4279  intiin  4356  dfint3  30504
  Copyright terms: Public domain W3C validator