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

Theorem uneq2 3336
Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
uneq2  |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )

Proof of Theorem uneq2
StepHypRef Expression
1 uneq1 3335 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
2 uncom 3332 . 2  |-  ( C  u.  A )  =  ( A  u.  C
)
3 uncom 3332 . 2  |-  ( C  u.  B )  =  ( B  u.  C
)
41, 2, 33eqtr4g 2353 1  |-  ( A  =  B  ->  ( C  u.  A )  =  ( C  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    u. cun 3163
This theorem is referenced by:  uneq12  3337  uneq2i  3339  uneq2d  3342  uneqin  3433  disjssun  3525  uniprg  3858  sucprc  4483  unexb  4536  undifixp  6868  unxpdom  7086  ackbij1lem16  7877  fin23lem28  7982  ttukeylem6  8157  ipodrsima  14284  mplsubglem  16195  mretopd  16845  iscldtop  16848  dfcon2  17161  nconsubb  17165  spanun  22140  nofulllem1  24427  brsuccf  24551  rankung  24868  domfldref  25164  comppfsc  26410  nacsfix  26890  eldioph4b  26997  eldioph4i  26998  fiuneneq  27616  paddval  30609  dochsatshp  32263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170
  Copyright terms: Public domain W3C validator