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Theorem uneq2 3627
Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem uneq2
StepHypRef Expression
1 uneq1 3626 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uncom 3623 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3623 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2573 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  cun 3442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-v 3079  df-un 3449
This theorem is referenced by:  uneq12  3628  uneq2i  3630  uneq2d  3633  uneqin  3740  disjssun  3891  uniprg  4284  unexb  6731  undifixp  7705  unxpdom  7927  ackbij1lem16  8815  fin23lem28  8920  ttukeylem6  9094  lcmfun  15070  ipodrsima  16878  mplsubglem  19157  mretopd  20607  iscldtop  20610  dfcon2  20933  nconsubb  20937  comppfsc  21046  spanun  27577  locfinref  29033  isros  29355  unelros  29358  difelros  29359  rossros  29367  inelcarsg  29507  nofulllem1  30937  rankung  31279  paddval  33977  dochsatshp  35633  nacsfix  36168  eldioph4b  36268  eldioph4i  36269  fiuneneq  36676  isotone1  37248  fiiuncl  38142
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