Theorem caovord3d

Description: Ordering law. (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 )
	caovord2d.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
	caovord3d.5	⊢ ( 𝜑 → 𝐷 ∈ 𝑆 )
Assertion	caovord3d	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 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		caovord2d.com	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) )
6		caovord3d.5	⊢ ( 𝜑 → 𝐷 ∈ 𝑆 )
7		breq1	⊢ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) → ( ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
8	1 2 4 3 5	caovord2d	⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
9	1 6 3 4	caovordd	⊢ ( 𝜑 → ( 𝐷 𝑅 𝐵 ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) )
10	8 9	bibi12d	⊢ ( 𝜑 → ( ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 𝐵 ) ↔ ( ( 𝐴 𝐹 𝐵 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ↔ ( 𝐶 𝐹 𝐷 ) 𝑅 ( 𝐶 𝐹 𝐵 ) ) ) )
11	7 10	syl5ibr	⊢ ( 𝜑 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐶 𝐹 𝐷 ) → ( 𝐴 𝑅 𝐶 ↔ 𝐷 𝑅 𝐵 ) ) )