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Theorem xpeq12 4558
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 4553 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2 xpeq2 4554 . 2  |-  ( C  =  D  ->  ( B  X.  C )  =  ( B  X.  D
) )
31, 2sylan9eq 2192 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    X. cxp 4537
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-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  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-opab 3990  df-xp 4545
This theorem is referenced by:  xpeq12i  4561  xpeq12d  4564  xpid11  4762  xp11m  4977  txtopon  12441  txbasval  12446  ismet  12523  isxmet  12524
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