Theorem caovordg

Description: Convert an operation ordering law to class notation. (Contributed by NM, 19-Feb-1996) (Revised by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypothesis	caovordg.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
Assertion	caovordg	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 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	1	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
3		breq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝑅 𝑦 ↔ 𝐴 𝑅 𝑦 ) )
4		oveq2	⊢ ( 𝑥 = 𝐴 → ( 𝑧 𝐹 𝑥 ) = ( 𝑧 𝐹 𝐴 ) )
5	4	breq1d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) )
6	3 5	bibi12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ↔ ( 𝐴 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ) )
7		breq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝑅 𝑦 ↔ 𝐴 𝑅 𝐵 ) )
8		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝑧 𝐹 𝑦 ) = ( 𝑧 𝐹 𝐵 ) )
9	8	breq2d	⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) )
10	7 9	bibi12d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) ↔ ( 𝐴 𝑅 𝐵 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) ) )
11		oveq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐹 𝐴 ) = ( 𝐶 𝐹 𝐴 ) )
12		oveq1	⊢ ( 𝑧 = 𝐶 → ( 𝑧 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐵 ) )
13	11 12	breq12d	⊢ ( 𝑧 = 𝐶 → ( ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
14	13	bibi2d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝑅 𝐵 ↔ ( 𝑧 𝐹 𝐴 ) 𝑅 ( 𝑧 𝐹 𝐵 ) ) ↔ ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) )
15	6 10 14	rspc3v	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( 𝑥 𝑅 𝑦 ↔ ( 𝑧 𝐹 𝑥 ) 𝑅 ( 𝑧 𝐹 𝑦 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) )
16	2 15	mpan9	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( 𝐴 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐴 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )