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Theorem xpeq12 4837
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 4832 . 2  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
2 xpeq2 4833 . 2  |-  ( C  =  D  ->  ( B  X.  C )  =  ( B  X.  D
) )
31, 2sylan9eq 2439 1  |-  ( ( A  =  B  /\  C  =  D )  ->  ( A  X.  C
)  =  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    X. cxp 4816
This theorem is referenced by:  xpeq12i  4840  xpeq12d  4843  xpid11  5031  xp11  5244  infxpenlem  7828  fpwwe2lem5  8442  pwfseqlem4a  8469  pwfseqlem4  8470  pwfseqlem5  8471  pwfseq  8472  pwsval  13635  txtopon  17544  txbasval  17559  txindislem  17586  ismet  18262  isxmet  18263  ismgm  21756  opidon2  21760  shsval  22662  prdsbnd2  26195  ttac  26798  mamufval  27112  sblpnf  27208
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-opab 4208  df-xp 4824
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