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Theorem uneq1 3501
Description: Equality theorem for union of two classes. (Contributed by NM, 15-Jul-1993.)
Assertion
Ref Expression
uneq1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )

Proof of Theorem uneq1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq2 2502 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21orbi1d 702 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  \/  x  e.  C
)  <->  ( x  e.  B  \/  x  e.  C ) ) )
3 elun 3495 . . 3  |-  ( x  e.  ( A  u.  C )  <->  ( x  e.  A  \/  x  e.  C ) )
4 elun 3495 . . 3  |-  ( x  e.  ( B  u.  C )  <->  ( x  e.  B  \/  x  e.  C ) )
52, 3, 43bitr4g 288 . 2  |-  ( A  =  B  ->  (
x  e.  ( A  u.  C )  <->  x  e.  ( B  u.  C
) ) )
65eqrdv 2439 1  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1369    e. wcel 1756    u. cun 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-un 3331
This theorem is referenced by:  uneq2  3502  uneq12  3503  uneq1i  3504  uneq1d  3507  unineq  3598  prprc1  3983  uniprg  4103  relresfld  5362  relcoi1  5364  unexb  6378  oarec  6999  xpider  7169  ralxpmap  7260  undifixp  7297  unxpdom  7518  enp1ilem  7544  findcard2  7550  domunfican  7582  pwfilem  7603  fin1a2lem10  8576  incexclem  13297  ramub1lem1  14085  ramub1  14087  mreexexlem3d  14582  mreexexlem4d  14583  ipodrsima  15333  mplsubglem  17508  mplsubglemOLD  17510  mretopd  18694  iscldtop  18697  nconsubb  19025  plyval  21659  spanun  24946  difeq  25897  measun  26623  nofulllem2  27842  brsuccf  27970  altopthsn  27990  rankung  28202  nacsfix  29045  eldioph4b  29147  eldioph4i  29148  diophren  29149  compne  29693  islshp  32621  lshpset2N  32761  paddval  33439
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