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Theorem uneq12 3740
 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 3738 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 3739 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2675 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∪ cun 3553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560 This theorem is referenced by:  uneq12i  3743  uneq12d  3746  un00  3983  opthprc  5127  dmpropg  5567  unixp  5627  fntpg  5906  fnun  5955  resasplit  6031  fvun  6225  rankprb  8658  pm54.43  8770  xpscg  16139  evlseu  19435  ptuncnv  21520  sshjval  28055  bj-2upleq  32644  poimirlem4  33042  poimirlem9  33047  diophun  36814  pwssplit4  37136  clsk1indlem3  37820
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