Metamath Proof Explorer

Theorem caov411d

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 ) ) )
caovd.4 
|- ( ph -> D e. S )
caovd.cl 
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
Assertion caov411d 
|- ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )

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 caovd.4 
 |-  ( ph -> D e. S )
7 caovd.cl 
 |-  ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )
8 2 1 3 4 5 6 7 caov4d 
 |-  ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( B F C ) F ( A F D ) ) )
9 4 2 1 caovcomd 
 |-  ( ph -> ( B F A ) = ( A F B ) )
10 9 oveq1d 
 |-  ( ph -> ( ( B F A ) F ( C F D ) ) = ( ( A F B ) F ( C F D ) ) )
11 4 2 3 caovcomd 
 |-  ( ph -> ( B F C ) = ( C F B ) )
12 11 oveq1d 
 |-  ( ph -> ( ( B F C ) F ( A F D ) ) = ( ( C F B ) F ( A F D ) ) )
13 8 10 12 3eqtr3d 
 |-  ( ph -> ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) )