Theorem caovcand

Description: Convert an operation cancellation law to class notation. (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 )
Assertion	caovcand	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 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		id	⊢ ( 𝜑 → 𝜑 )
6	1	caovcang	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) )
7	5 2 3 4 6	syl13anc	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) )