Step |
Hyp |
Ref |
Expression |
1 |
|
caov.1 |
|- A e. _V |
2 |
|
caov.2 |
|- B e. _V |
3 |
|
caov.3 |
|- C e. _V |
4 |
|
caov.com |
|- ( x F y ) = ( y F x ) |
5 |
|
caov.ass |
|- ( ( x F y ) F z ) = ( x F ( y F z ) ) |
6 |
|
caov.4 |
|- D e. _V |
7 |
1 2 3 4 5
|
caov31 |
|- ( ( A F B ) F C ) = ( ( C F B ) F A ) |
8 |
7
|
oveq1i |
|- ( ( ( A F B ) F C ) F D ) = ( ( ( C F B ) F A ) F D ) |
9 |
|
ovex |
|- ( A F B ) e. _V |
10 |
9 3 6 5
|
caovass |
|- ( ( ( A F B ) F C ) F D ) = ( ( A F B ) F ( C F D ) ) |
11 |
|
ovex |
|- ( C F B ) e. _V |
12 |
11 1 6 5
|
caovass |
|- ( ( ( C F B ) F A ) F D ) = ( ( C F B ) F ( A F D ) ) |
13 |
8 10 12
|
3eqtr3i |
|- ( ( A F B ) F ( C F D ) ) = ( ( C F B ) F ( A F D ) ) |