Metamath Proof Explorer
Description: Convert an operation commutative law to class notation. (Contributed by NM, 26-Aug-1995) (Revised by Mario Carneiro, 1-Jun-2013)
|
|
Ref |
Expression |
|
Hypotheses |
caovcom.1 |
⊢ 𝐴 ∈ V |
|
|
caovcom.2 |
⊢ 𝐵 ∈ V |
|
|
caovcom.3 |
⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) |
|
Assertion |
caovcom |
⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
caovcom.1 |
⊢ 𝐴 ∈ V |
2 |
|
caovcom.2 |
⊢ 𝐵 ∈ V |
3 |
|
caovcom.3 |
⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) |
4 |
1 2
|
pm3.2i |
⊢ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) |
5 |
3
|
a1i |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) ) |
6 |
5
|
caovcomg |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) |
7 |
1 4 6
|
mp2an |
⊢ ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) |