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Theorem altxpeq2 24580
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 2750 . . . 4  |-  ( A  =  B  ->  ( E. y  e.  A  z  =  << x ,  y >> 
<->  E. y  e.  B  z  =  << x ,  y >> ) )
21rexbidv 2577 . . 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 2410 . 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 24565 . 2  |-  ( C 
XX.  A )  =  { z  |  E. x  e.  C  E. y  e.  A  z  =  << x ,  y
>> }
5 df-altxp 24565 . 2  |-  ( C 
XX.  B )  =  { z  |  E. x  e.  C  E. y  e.  B  z  =  << x ,  y
>> }
63, 4, 53eqtr4g 2353 1  |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   {cab 2282   E.wrex 2557   <<caltop 24562    XX. caltxp 24563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-altxp 24565
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