Theorem caovcang

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995) (Revised by Mario Carneiro, 30-Dec-2014)

	Ref	Expression
Hypothesis	caovcang.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
Assertion	caovcang	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 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	1	ralrimivvva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝑇 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) )
3		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑦 ) )
4		oveq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 𝐹 𝑧 ) = ( 𝐴 𝐹 𝑧 ) )
5	3 4	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑧 ) ) )
6	5	bibi1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) ↔ ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) ) )
7		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
8	7	eqeq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑧 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝑧 ) ) )
9		eqeq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝑧 ↔ 𝐵 = 𝑧 ) )
10	8 9	bibi12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) ↔ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝑧 ) ↔ 𝐵 = 𝑧 ) ) )
11		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝐴 𝐹 𝑧 ) = ( 𝐴 𝐹 𝐶 ) )
12	11	eqeq2d	⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝑧 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ) )
13		eqeq2	⊢ ( 𝑧 = 𝐶 → ( 𝐵 = 𝑧 ↔ 𝐵 = 𝐶 ) )
14	12 13	bibi12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝑧 ) ↔ 𝐵 = 𝑧 ) ↔ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) ) )
15	6 10 14	rspc3v	⊢ ( ( 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) → ( ∀ 𝑥 ∈ 𝑇 ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ 𝑦 = 𝑧 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) ) )
16	2 15	mpan9	⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ) ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ↔ 𝐵 = 𝐶 ) )