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Theorem unieq 3589
 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

Proof of Theorem unieq
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2506 . . 3
21abbidv 2155 . 2
3 dfuni2 3582 . 2
4 dfuni2 3582 . 2
52, 3, 43eqtr4g 2097 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1243  cab 2026  wrex 2307  cuni 3580 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-uni 3581 This theorem is referenced by:  unieqi  3590  unieqd  3591  uniintsnr  3651  iununir  3738  treq  3860  limeq  4114  uniex  4174  uniexg  4175  ordsucunielexmid  4256  onsucuni2  4288  elvvuni  4404  unielrel  4845  unixp0im  4854  iotass  4884  en1bg  6280  bj-uniex  10037  bj-uniexg  10038
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