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Theorem inteq 3810
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 2652 . . 3  |-  ( A  =  B  ->  ( A. y  e.  A  x  e.  y  <->  A. y  e.  B  x  e.  y ) )
21abbidv 2275 . 2  |-  ( A  =  B  ->  { x  |  A. y  e.  A  x  e.  y }  =  { x  |  A. y  e.  B  x  e.  y } )
3 dfint2 3809 . 2  |-  |^| A  =  { x  |  A. y  e.  A  x  e.  y }
4 dfint2 3809 . 2  |-  |^| B  =  { x  |  A. y  e.  B  x  e.  y }
52, 3, 43eqtr4g 2215 1  |-  ( A  =  B  ->  |^| A  =  |^| B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335   {cab 2143   A.wral 2435   |^|cint 3807
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-int 3808
This theorem is referenced by:  inteqi  3811  inteqd  3812  uniintsnr  3843  rint0  3846  intexr  4111  onintexmid  4531  elreldm  4811  elxp5  5073  1stval2  6100  fundmen  6748  xpsnen  6763  fiintim  6870  elfir  6914  fiinopn  12389  bj-intexr  13470
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