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 ) )