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Theorem iuneq2i 4809
Description: Equality inference for indexed union. (Contributed by NM, 22-Oct-2003.)
Hypothesis
Ref Expression
iuneq2i.1 (𝑥𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iuneq2i 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶

Proof of Theorem iuneq2i
StepHypRef Expression
1 iuneq2 4807 . 2 (∀𝑥𝐴 𝐵 = 𝐶 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶)
2 iuneq2i.1 . 2 (𝑥𝐴𝐵 = 𝐶)
31, 2mprg 3097 1 𝑥𝐴 𝐵 = 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  wcel 2051   ciun 4789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-v 3412  df-in 3831  df-ss 3838  df-iun 4791
This theorem is referenced by:  dfiunv2  4827  iunrab  4839  iunid  4847  iunin1  4857  2iunin  4861  resiun1  5716  resiun2  5717  dfimafn2  6557  dfmpt  6728  funiunfv  6831  fpar  7618  onovuni  7782  uniqs  8156  marypha2lem2  8694  alephlim  9286  cfsmolem  9489  ituniiun  9641  imasdsval2  16644  lpival  19752  cmpsublem  21727  txbasval  21934  uniioombllem2  23903  uniioombllem4  23906  volsup2  23925  itg1addlem5  24020  itg1climres  24034  indval2  30950  sigaclfu2  31058  measvuni  31151  fmla  32224  trpred0  32629  rabiun  34339  mblfinlem2  34404  voliunnfl  34410  cnambfre  34414  uniqsALTV  35064  trclrelexplem  39453  cotrclrcl  39484  dfcoll2  39997  hoicvr  42291  hoidmv1le  42337  hoidmvle  42343  hspmbllem2  42370  smflimlem3  42510  smflimlem4  42511  smflim  42514  dfaimafn2  42801  xpiun  43431
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