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Theorem uneq12 3456
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 3454 . 2  |-  ( A  =  B  ->  ( A  u.  C )  =  ( B  u.  C ) )
2 uneq2 3455 . 2  |-  ( C  =  D  ->  ( B  u.  C )  =  ( B  u.  D ) )
31, 2sylan9eq 2456 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  u.  C
)  =  ( B  u.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    u. cun 3278
This theorem is referenced by:  uneq12i  3459  uneq12d  3462  un00  3623  opthprc  4884  dmpropg  5302  unixp  5361  fntpg  5465  fnun  5510  resasplit  5572  fvun  5752  rankprb  7733  pm54.43  7843  xpscg  13738  ptuncnv  17792  evlseu  19890  sshjval  22805  diophun  26722  pwssplit4  27059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285
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