Step |
Hyp |
Ref |
Expression |
1 |
|
caovd.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑆 ) |
2 |
|
caovd.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑆 ) |
3 |
|
caovd.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑆 ) |
4 |
|
caovd.com |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) ) |
5 |
|
caovd.ass |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) ) |
6 |
4 1 3
|
caovcomd |
⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐶 𝐹 𝐴 ) ) |
7 |
6
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) = ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) ) |
8 |
1 2 3 4 5
|
caov32d |
⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐴 𝐹 𝐶 ) 𝐹 𝐵 ) ) |
9 |
3 2 1 4 5
|
caov32d |
⊢ ( 𝜑 → ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 ) = ( ( 𝐶 𝐹 𝐴 ) 𝐹 𝐵 ) ) |
10 |
7 8 9
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) 𝐹 𝐶 ) = ( ( 𝐶 𝐹 𝐵 ) 𝐹 𝐴 ) ) |