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Theorem altxpeq2 25784
Description: Equality for alternate cross products. (Contributed by Scott Fenton, 24-Mar-2012.)
Assertion
Ref Expression
altxpeq2  |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B
) )

Proof of Theorem altxpeq2
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexeq 2897 . . . 4  |-  ( A  =  B  ->  ( E. y  e.  A  z  =  << x ,  y >> 
<->  E. y  e.  B  z  =  << x ,  y >> ) )
21rexbidv 2718 . . 3  |-  ( A  =  B  ->  ( E. x  e.  C  E. y  e.  A  z  =  << x ,  y >> 
<->  E. x  e.  C  E. y  e.  B  z  =  << x ,  y >> ) )
32abbidv 2549 . 2  |-  ( A  =  B  ->  { z  |  E. x  e.  C  E. y  e.  A  z  =  << x ,  y >> }  =  { z  |  E. x  e.  C  E. y  e.  B  z  =  << x ,  y
>> } )
4 df-altxp 25769 . 2  |-  ( C 
XX.  A )  =  { z  |  E. x  e.  C  E. y  e.  A  z  =  << x ,  y
>> }
5 df-altxp 25769 . 2  |-  ( C 
XX.  B )  =  { z  |  E. x  e.  C  E. y  e.  B  z  =  << x ,  y
>> }
63, 4, 53eqtr4g 2492 1  |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   {cab 2421   E.wrex 2698   <<caltop 25766    XX. caltxp 25767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-altxp 25769
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