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Theorem uneq12 3285
Description: Equality theorem for union of two classes. (Contributed by NM, 29-Mar-1998.)
Assertion
Ref Expression
uneq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )

Proof of Theorem uneq12
StepHypRef Expression
1 uneq1 3283 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
2 uneq2 3284 . 2  |-  ( C  =  D  ->  ( B  u.  C )  =  ( B  u.  D ) )
31, 2sylan9eq 2308 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    u. cun 3111
This theorem is referenced by:  uneq12i  3288  uneq12d  3291  un00  3451  opthprc  4710  dmpropg  5119  unixp  5178  fnun  5274  resasplit  5335  fvun  5509  rankprb  7477  pm54.43  7587  xpscg  13408  ptuncnv  17446  evlseu  19348  sshjval  21875  hdrmp  25059  diophun  26206  pwssplit4  26544
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118
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