Theorem caovcomd

Description: Convert an operation commutative law to class notation. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovcomg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
	caovcomd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovcomd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
Assertion	caovcomd	⊢ ( 𝜑 → ( 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		caovcomd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3		caovcomd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		id	⊢ ( 𝜑 → 𝜑 )
5	1	caovcomg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) ) → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )
6	4 2 3 5	syl12anc	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )