Metamath Proof Explorer


Theorem caov32d

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 caov32d
|- ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )

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 2 3 caovcomd
 |-  ( ph -> ( B F C ) = ( C F B ) )
7 6 oveq2d
 |-  ( ph -> ( A F ( B F C ) ) = ( A F ( C F B ) ) )
8 5 1 2 3 caovassd
 |-  ( ph -> ( ( A F B ) F C ) = ( A F ( B F C ) ) )
9 5 1 3 2 caovassd
 |-  ( ph -> ( ( A F C ) F B ) = ( A F ( C F B ) ) )
10 7 8 9 3eqtr4d
 |-  ( ph -> ( ( A F B ) F C ) = ( ( A F C ) F B ) )