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Theorem uneq12 4131
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 4129 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 4130 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2873 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  cun 3931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-un 3938
This theorem is referenced by:  uneq12i  4134  uneq12d  4137  un00  4390  opthprc  5609  dmpropg  6065  unixp  6126  fntpg  6407  fnun  6456  resasplit  6541  fvun  6746  rankprb  9268  pm54.43  9417  pwmndgplus  18038  evlseu  20224  ptuncnv  22343  sshjval  29054  bj-2upleq  34221  poimirlem4  34777  poimirlem9  34782  diophun  39248  pwssplit4  39567  clsk1indlem3  40271
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