Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995)
Ref | Expression | ||
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Hypotheses | caov.1 | |- A e. _V |
|
caov.2 | |- B e. _V |
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caov.3 | |- C e. _V |
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caov.com | |- ( x F y ) = ( y F x ) |
||
caov.ass | |- ( ( x F y ) F z ) = ( x F ( y F z ) ) |
||
Assertion | caov13 | |- ( A F ( B F C ) ) = ( C F ( B F A ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | |- A e. _V |
|
2 | caov.2 | |- B e. _V |
|
3 | caov.3 | |- C e. _V |
|
4 | caov.com | |- ( x F y ) = ( y F x ) |
|
5 | caov.ass | |- ( ( x F y ) F z ) = ( x F ( y F z ) ) |
|
6 | 1 2 3 4 5 | caov31 | |- ( ( A F B ) F C ) = ( ( C F B ) F A ) |
7 | 1 2 3 5 | caovass | |- ( ( A F B ) F C ) = ( A F ( B F C ) ) |
8 | 3 2 1 5 | caovass | |- ( ( C F B ) F A ) = ( C F ( B F A ) ) |
9 | 6 7 8 | 3eqtr3i | |- ( A F ( B F C ) ) = ( C F ( B F A ) ) |