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Theorem inteq 3609
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 2499 . . 3 (A = B → (y A x yy B x y))
21abbidv 2152 . 2 (A = B → {xy A x y} = {xy B x y})
3 dfint2 3608 . 2 A = {xy A x y}
4 dfint2 3608 . 2 B = {xy B x y}
52, 3, 43eqtr4g 2094 1 (A = B A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  {cab 2023  wral 2300   cint 3606
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-int 3607
This theorem is referenced by:  inteqi  3610  inteqd  3611  uniintsnr  3642  rint0  3645  intexr  3895  elreldm  4503  elxp5  4752  1stval2  5724  fundmen  6222  xpsnen  6231  bj-intexr  9363
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