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Theorem xpeq12 4420
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 4415 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2 xpeq2 4416 . 2  |-  ( C  =  D  ->  ( B  X.  C )  =  ( B  X.  D
) )
31, 2sylan9eq 2135 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    X. cxp 4399
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-opab 3866  df-xp 4407
This theorem is referenced by:  xpeq12i  4423  xpeq12d  4426  xpid11m  4616  xp11m  4823
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