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Theorem xpeq1 4302
Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
xpeq1  X.  C  X.  C

Proof of Theorem xpeq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2098 . . . 4
21anbi1d 438 . . 3  C  C
32opabbidv 3814 . 2  { <. ,  >.  |  C }  { <. ,  >.  |  C }
4 df-xp 4294 . 2  X.  C  { <. , 
>.  |  C }
5 df-xp 4294 . 2  X.  C  { <. , 
>.  |  C }
63, 4, 53eqtr4g 2094 1  X.  C  X.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {copab 3808    X. cxp 4286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-opab 3810  df-xp 4294
This theorem is referenced by:  xpeq12  4307  xpeq1i  4308  xpeq1d  4311  opthprc  4334  reseq2  4550  xpeq0r  4689  xpdisj1  4690  xpima1  4710  xpsneng  6232  xpcomeng  6238  xpdom2g  6242
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