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Theorem unieq 3563
Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
unieq (A = B A = B)

Proof of Theorem unieq
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2484 . . 3 (A = B → (x A y xx B y x))
21abbidv 2137 . 2 (A = B → {yx A y x} = {yx B y x})
3 dfuni2 3556 . 2 A = {yx A y x}
4 dfuni2 3556 . 2 B = {yx B y x}
52, 3, 43eqtr4g 2079 1 (A = B A = B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  {cab 2008  wrex 2285   cuni 3554
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-uni 3555
This theorem is referenced by:  unieqi  3564  unieqd  3565  uniintsnr  3625  iununir  3712  treq  3834  limeq  4063  uniex  4124  uniexg  4125  ordsucunielexmid  4200  elvvuni  4331  unielrel  4772  unixp0im  4781  iotass  4811  bj-uniex  7140  bj-uniexg  7141
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