Theorem caovordd

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

	Ref	Expression
Hypotheses	caovordg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
	caovordd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
	caovordd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
	caovordd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
Assertion	caovordd	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		caovordg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
2		caovordd.2	⊢ ( 𝜑 → 𝐴 ∈ 𝑆 )
3		caovordd.3	⊢ ( 𝜑 → 𝐵 ∈ 𝑆 )
4		caovordd.4	⊢ ( 𝜑 → 𝐶 ∈ 𝑆 )
5		id	⊢ ( 𝜑 → 𝜑 )
6	1	caovordg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
7	5 2 3 4 6	syl13anc	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )