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Theorem uneq12 4132
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 4130 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 4131 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2874 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  cun 3932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-un 3939
This theorem is referenced by:  uneq12i  4135  uneq12d  4138  un00  4392  opthprc  5609  dmpropg  6065  unixp  6126  fntpg  6407  fnun  6456  resasplit  6541  fvun  6746  rankprb  9272  pm54.43  9421  pwmndgplus  18092  evlseu  20288  ptuncnv  22407  sshjval  29119  bj-2upleq  34312  poimirlem4  34878  poimirlem9  34883  diophun  39350  pwssplit4  39669  clsk1indlem3  40373
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