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Theorem inteq 3881
Description: Equality law for intersection. (Contributed by NM, 13-Sep-1999.)
Assertion
Ref Expression
inteq  |-  ( A  =  B  ->  |^| A  =  |^| B )

Proof of Theorem inteq
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 2749 . . 3  |-  ( A  =  B  ->  ( A. y  e.  A  x  e.  y  <->  A. y  e.  B  x  e.  y ) )
21abbidv 2410 . 2  |-  ( A  =  B  ->  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y  e.  B  x  e.  y } )
3 dfint2 3880 . 2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
4 dfint2 3880 . 2  |-  |^| B  =  { x  |  A. y  e.  B  x  e.  y }
52, 3, 43eqtr4g 2353 1  |-  ( A  =  B  ->  |^| A  =  |^| B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   |^|cint 3878
This theorem is referenced by:  inteqi  3882  inteqd  3883  unissint  3902  uniintsn  3915  rint0  3918  intex  4183  intnex  4184  elreldm  4919  elxp5  5177  1stval2  6153  oev2  6538  fundmen  6950  xpsnen  6962  fiint  7149  elfir  7185  fiin  7191  cardmin2  7647  isfin2-2  7961  incexclem  12311  incexc  12312  xpnnenOLD  12504  mreintcl  13513  ismred2  13521  fiinopn  16663  cmpfii  17152  ptbasfi  17292  fbssint  17549  shintcl  21925  chintcl  21927  rankeq1o  24873  isconcl2b  26201  neificl  26570  heibor1lem  26636  elrfi  26872  elrfirn  26873
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-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-int 3879
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