Theorem xpeq12 4639

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

Step	Hyp	Ref	Expression
1		xpeq1 4634	. 2 |- ( A = B -> ( A X. C ) = ( B X. C ) )
2		xpeq2 4635	. 2 |- ( C = D -> ( B X. C ) = ( B X. D ) )
3	1, 2	sylan9eq 2228	1 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   X. cxp 4618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-11 1504  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-opab 4060  df-xp 4626

This theorem is referenced by:  xpeq12i  4642  xpeq12d  4645  xpid11  4843  xp11m  5059  txtopon  13333  txbasval  13338  ismet  13415  isxmet  13416