Step |
Hyp |
Ref |
Expression |
1 |
|
caov.1 |
⊢ 𝐴 ∈ V |
2 |
|
caov.2 |
⊢ 𝐵 ∈ V |
3 |
|
caov.3 |
⊢ 𝐶 ∈ 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 ∈ 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 ) ∈ 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 ) ∈ 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 ) ) |