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Theorem inteq 3834
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 2665 . . 3  |-  ( A  =  B  ->  ( A. y  e.  A  x  e.  y  <->  A. y  e.  B  x  e.  y ) )
21abbidv 2288 . 2  |-  ( A  =  B  ->  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y  e.  B  x  e.  y } )
3 dfint2 3833 . 2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
4 dfint2 3833 . 2  |-  |^| B  =  { x  |  A. y  e.  B  x  e.  y }
52, 3, 43eqtr4g 2228 1  |-  ( A  =  B  ->  |^| A  =  |^| B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   {cab 2156   A.wral 2448   |^|cint 3831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-int 3832
This theorem is referenced by:  inteqi  3835  inteqd  3836  uniintsnr  3867  rint0  3870  intexr  4136  onintexmid  4557  elreldm  4837  elxp5  5099  1stval2  6134  fundmen  6784  xpsnen  6799  fiintim  6906  elfir  6950  fiinopn  12796  bj-intexr  13943
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