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Theorem xpeq12 4889
Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
xpeq12  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )

Proof of Theorem xpeq12
StepHypRef Expression
1 xpeq1 4884 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2 xpeq2 4885 . 2  |-  ( C  =  D  ->  ( B  X.  C )  =  ( B  X.  D
) )
31, 2sylan9eq 2487 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    X. cxp 4868
This theorem is referenced by:  xpeq12i  4892  xpeq12d  4895  xpid11  5083  xp11  5296  infxpenlem  7887  fpwwe2lem5  8501  pwfseqlem4a  8528  pwfseqlem4  8529  pwfseqlem5  8530  pwfseq  8531  pwsval  13700  txtopon  17615  txbasval  17630  txindislem  17657  ismet  18345  isxmet  18346  ismgm  21900  opidon2  21904  shsval  22806  prdsbnd2  26495  ttac  27098  mamufval  27411  sblpnf  27507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-opab 4259  df-xp 4876
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