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Metamath Proof Explorer

Theorem caov31d

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

Ref Expression
Hypotheses caovd.1 
|- ( ph -> A e. S )
caovd.2 
|- ( ph -> B e. S )
caovd.3 
|- ( ph -> C e. S )
caovd.com 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
caovd.ass 
|- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
Assertion caov31d 
|- ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )

Proof

Step Hyp Ref Expression
1 caovd.1 
 |-  ( ph -> A e. S )
2 caovd.2 
 |-  ( ph -> B e. S )
3 caovd.3 
 |-  ( ph -> C e. S )
4 caovd.com 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )
5 caovd.ass 
 |-  ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )
6 4 1 3 caovcomd 
 |-  ( ph -> ( A F C ) = ( C F A ) )
7 6 oveq1d 
 |-  ( ph -> ( ( A F C ) F B ) = ( ( C F A ) F B ) )
8 1 2 3 4 5 caov32d 
 |-  ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )
9 3 2 1 4 5 caov32d 
 |-  ( ph -> ( ( C F B ) F A ) = ( ( C F A ) F B ) )
10 7 8 9 3eqtr4d 
 |-  ( ph -> ( ( A F B ) F C ) = ( ( C F B ) F A ) )