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Theorem uneq12 3147
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 3145 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
2 uneq2 3146 . 2  |-  ( C  =  D  ->  ( B  u.  C )  =  ( B  u.  D ) )
31, 2sylan9eq 2140 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    u. cun 2995
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001
This theorem is referenced by:  uneq12i  3150  uneq12d  3153  un00  3326  opthprc  4477  dmpropg  4890  unixpm  4953  fntpg  5056  fnun  5106  resasplitss  5174  pm54.43  6797
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