Theorem caovcanrd

Description: Commute the arguments of an operation cancellation law. (Contributed by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypotheses	caovcang.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
	caovcand.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑇 )
	caovcand.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovcand.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
	caovcanrd.5	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovcanrd.6	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
Assertion	caovcanrd	⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ↔ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		caovcang.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
2		caovcand.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑇 )
3		caovcand.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		caovcand.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5		caovcanrd.5	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
6		caovcanrd.6	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
7	6 5 3	caovcomd	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐵 ) = ( 𝐵 𝐹 𝐴 ) )
8	6 5 4	caovcomd	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐶 𝐹 𝐴 ) )
9	7 8	eqeq12d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ) )
10	1 2 3 4	caovcand	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) )
11	9 10	bitr3d	⊢ ( 𝜑 → ( ( 𝐵 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) ↔ 𝐵 = 𝐶 ) )