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Theorem uneq12 3225
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 3223 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
2 uneq2 3224 . 2  |-  ( C  =  D  ->  ( B  u.  C )  =  ( B  u.  D ) )
31, 2sylan9eq 2192 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    u. cun 3069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075
This theorem is referenced by:  uneq12i  3228  uneq12d  3231  un00  3409  opthprc  4590  dmpropg  5011  unixpm  5074  fntpg  5179  fnun  5229  resasplitss  5302  pm54.43  7051
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