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Theorem uneq12 3724
Description: Equality theorem for the union of two classes. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
uneq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem uneq12
StepHypRef Expression
1 uneq1 3722 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 3723 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2664 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  cun 3538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545
This theorem is referenced by:  uneq12i  3727  uneq12d  3730  un00  3963  opthprc  5089  dmpropg  5526  unixp  5585  fntpg  5862  fnun  5911  resasplit  5987  fvun  6178  rankprb  8597  pm54.43  8709  xpscg  16041  evlseu  19337  ptuncnv  21420  sshjval  27593  bj-2upleq  32193  poimirlem4  32583  poimirlem9  32588  diophun  36355  pwssplit4  36677  clsk1indlem3  37361
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