Theorem caovcan

Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995)

	Ref	Expression
Hypotheses	caovcan.1	⊢ 𝐶 ∈ V
	caovcan.2	⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) )
Assertion	caovcan	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		caovcan.1	⊢ 𝐶 ∈ V
2		caovcan.2	⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 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 𝐶 ) = ( 𝐴 𝐹 𝐶 ) )
5	3 4	eqeq12d	⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ) )
6	5	imbi1d	⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) )
7		oveq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐵 ) )
8	7	eqeq1d	⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) ↔ ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) ) )
9		eqeq1	⊢ ( 𝑦 = 𝐵 → ( 𝑦 = 𝐶 ↔ 𝐵 = 𝐶 ) )
10	8 9	imbi12d	⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 𝐹 𝑦 ) = ( 𝐴 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ↔ ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) ) )
11		oveq2	⊢ ( 𝑧 = 𝐶 → ( 𝑥 𝐹 𝑧 ) = ( 𝑥 𝐹 𝐶 ) )
12	11	eqeq2d	⊢ ( 𝑧 = 𝐶 → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) ↔ ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) ) )
13		eqeq2	⊢ ( 𝑧 = 𝐶 → ( 𝑦 = 𝑧 ↔ 𝑦 = 𝐶 ) )
14	12 13	imbi12d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ↔ ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) )
15	14	imbi2d	⊢ ( 𝑧 = 𝐶 → ( ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝑧 ) → 𝑦 = 𝑧 ) ) ↔ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) ) ) )
16	1 15 2	vtocl	⊢ ( ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) → ( ( 𝑥 𝐹 𝑦 ) = ( 𝑥 𝐹 𝐶 ) → 𝑦 = 𝐶 ) )
17	6 10 16	vtocl2ga	⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ) → ( ( 𝐴 𝐹 𝐵 ) = ( 𝐴 𝐹 𝐶 ) → 𝐵 = 𝐶 ) )