Metamath Proof Explorer


Theorem caov12d

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 caov12d
|- ( ph -> ( A F ( B F C ) ) = ( B F ( A F C ) ) )

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 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 ) ) )