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Theorem altxpeq2 25535
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 2850 . . . 4  |-  ( A  =  B  ->  ( E. y  e.  A  z  =  << x ,  y >> 
<->  E. y  e.  B  z  =  << x ,  y >> ) )
21rexbidv 2672 . . 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 2503 . 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 25520 . 2  |-  ( C 
XX.  A )  =  { z  |  E. x  e.  C  E. y  e.  A  z  =  << x ,  y
>> }
5 df-altxp 25520 . 2  |-  ( C 
XX.  B )  =  { z  |  E. x  e.  C  E. y  e.  B  z  =  << x ,  y
>> }
63, 4, 53eqtr4g 2446 1  |-  ( A  =  B  ->  ( C  XX.  A )  =  ( C  XX.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   {cab 2375   E.wrex 2652   <<caltop 25517    XX. caltxp 25518
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 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-rex 2657  df-altxp 25520
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