MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uneq2 Structured version   Visualization version   GIF version

Theorem uneq2 3745
Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem uneq2
StepHypRef Expression
1 uneq1 3744 . 2 (𝐴 = 𝐵 → (𝐴𝐶) = (𝐵𝐶))
2 uncom 3741 . 2 (𝐶𝐴) = (𝐴𝐶)
3 uncom 3741 . 2 (𝐶𝐵) = (𝐵𝐶)
41, 2, 33eqtr4g 2680 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  cun 3558
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 3192  df-un 3565
This theorem is referenced by:  uneq12  3746  uneq2i  3748  uneq2d  3751  uneqin  3860  disjssun  4014  uniprg  4423  unexb  6923  undifixp  7904  unxpdom  8127  ackbij1lem16  9017  fin23lem28  9122  ttukeylem6  9296  lcmfun  15301  ipodrsima  17105  mplsubglem  19374  mretopd  20836  iscldtop  20839  dfconn2  21162  nconnsubb  21166  comppfsc  21275  spanun  28292  locfinref  29732  isros  30054  unelros  30057  difelros  30058  rossros  30066  inelcarsg  30196  rankung  31968  paddval  34603  dochsatshp  36259  nacsfix  36794  eldioph4b  36894  eldioph4i  36895  fiuneneq  37295  isotone1  37867  fiiuncl  38756
  Copyright terms: Public domain W3C validator