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Theorem unieq 3754
 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 2631 . . 3 (𝐴 = 𝐵 → (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐵 𝑦𝑥))
21abbidv 2258 . 2 (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥} = {𝑦 ∣ ∃𝑥𝐵 𝑦𝑥})
3 dfuni2 3747 . 2 𝐴 = {𝑦 ∣ ∃𝑥𝐴 𝑦𝑥}
4 dfuni2 3747 . 2 𝐵 = {𝑦 ∣ ∃𝑥𝐵 𝑦𝑥}
52, 3, 43eqtr4g 2198 1 (𝐴 = 𝐵 𝐴 = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332  {cab 2126  ∃wrex 2418  ∪ cuni 3745 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-uni 3746 This theorem is referenced by:  unieqi  3755  unieqd  3756  uniintsnr  3816  iununir  3905  treq  4041  limeq  4308  uniex  4368  uniexg  4370  ordsucunielexmid  4455  onsucuni2  4488  nnpredcl  4545  elvvuni  4612  unielrel  5075  unixp0im  5084  iotass  5114  nnsucuniel  6400  en1bg  6703  omp1eom  6990  ctmlemr  7003  uniopn  12227  istopon  12239  eltg3  12285  tgdom  12300  cldval  12327  ntrfval  12328  clsfval  12329  neifval  12368  tgrest  12397  cnprcl2k  12434  bj-uniex  13306  bj-uniexg  13307  nnsf  13393  peano3nninf  13395  exmidsbthr  13412
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