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Metamath Proof Explorer

Theorem caovcomg

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 1-Jun-2013)

Ref Expression
Hypothesis caovcomg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
Assertion caovcomg ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )

Proof

Step Hyp Ref Expression
1 caovcomg.1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
2 1 ralrimivva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
3 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) )
4 oveq2 ⊢ ( 𝑥 = 𝐴 → ( 𝑦 𝐹 𝑥 ) = ( 𝑦 𝐹 𝐴 ) )
5 3 4 eqeq12d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) ↔ ( 𝐴 𝐹 𝑦 ) = ( 𝑦 𝐹 𝐴 ) ) )
6 oveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
7 oveq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 𝐹 𝐴 ) = ( 𝐵 𝐹 𝐴 ) )
8 6 7 eqeq12d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) = ( 𝑦 𝐹 𝐴 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) )
9 5 8 rspc2v ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) ) )
10 2 9 mpan9 ⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )