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 |
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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 ) |
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caov.ass | |- ( ( x F y ) F z ) = ( x F ( y F z ) ) |
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Assertion | caov32 | |- ( ( A F B ) F C ) = ( ( A F C ) F B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caov.1 | |- A e. _V |
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2 | caov.2 | |- B e. _V |
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3 | caov.3 | |- C e. _V |
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4 | caov.com | |- ( x F y ) = ( y F x ) |
|
5 | caov.ass | |- ( ( x F y ) F z ) = ( x F ( y F z ) ) |
|
6 | 2 3 4 | caovcom | |- ( B F C ) = ( C F B ) |
7 | 6 | oveq2i | |- ( A F ( B F C ) ) = ( A F ( C F B ) ) |
8 | 1 2 3 5 | caovass | |- ( ( A F B ) F C ) = ( A F ( B F C ) ) |
9 | 1 3 2 5 | caovass | |- ( ( A F C ) F B ) = ( A F ( C F B ) ) |
10 | 7 8 9 | 3eqtr4i | |- ( ( A F B ) F C ) = ( ( A F C ) F B ) |