Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)
Ref | Expression | ||
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Hypotheses | caovd.1 | |- ( ph -> A e. S ) |
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caovd.2 | |- ( ph -> B e. S ) |
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caovd.3 | |- ( ph -> C e. S ) |
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caovd.com | |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) ) |
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caovd.ass | |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) ) |
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Assertion | caov12d | |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) ) |
Step | Hyp | Ref | Expression |
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1 | caovd.1 | |- ( ph -> A e. S ) |
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2 | caovd.2 | |- ( ph -> B e. S ) |
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3 | caovd.3 | |- ( ph -> C e. S ) |
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4 | caovd.com | |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) ) |
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5 | caovd.ass | |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) ) |
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6 | 4 1 2 | caovcomd | |- ( ph -> ( A F B ) = ( B F A ) ) |
7 | 6 | oveq1d | |- ( ph -> ( ( A F B ) F C ) = ( ( B F A ) F C ) ) |
8 | 5 1 2 3 | caovassd | |- ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) ) |
9 | 5 2 1 3 | caovassd | |- ( ph -> ( ( B F A ) F C ) = ( B F ( A F C ) ) ) |
10 | 7 8 9 | 3eqtr3d | |- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) ) |