MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uneq12 Structured version   Visualization version   GIF version

Theorem uneq12 3913
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 3911 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uneq2 3912 . 2 (𝐶 = 𝐷 → (𝐵𝐶) = (𝐵𝐷))
31, 2sylan9eq 2825 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  cun 3721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728
This theorem is referenced by:  uneq12i  3916  uneq12d  3919  un00  4155  opthprc  5307  dmpropg  5750  unixp  5812  fntpg  6089  fnun  6137  resasplit  6214  fvun  6410  rankprb  8878  pm54.43  9026  xpscg  16426  evlseu  19731  ptuncnv  21831  sshjval  28549  bj-2upleq  33331  poimirlem4  33746  poimirlem9  33751  diophun  37863  pwssplit4  38185  clsk1indlem3  38867
  Copyright terms: Public domain W3C validator