Theorem caovcang 7351

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

Hypothesis

Ref	Expression
caovcang.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))

Assertion

Ref	Expression
caovcang	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝑇,𝑦,𝑧

Proof of Theorem caovcang

Step	Hyp	Ref	Expression
1		caovcang.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
2	1	ralrimivvva 3194	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
3		oveq1 7165	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4		oveq1 7165	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑧) = (𝐴𝐹𝑧))
5	3, 4	eqeq12d 2839	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝑧)))
6	5	bibi1d 346	. . 3 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧)))
7		oveq2 7166	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
8	7	eqeq1d 2825	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝑧)))
9		eqeq1 2827	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝑧 ↔ 𝐵 = 𝑧))
10	8, 9	bibi12d 348	. . 3 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧)))
11		oveq2 7166	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝐴𝐹𝑧) = (𝐴𝐹𝐶))
12	11	eqeq2d 2834	. . . 4 ⊢ (𝑧 = 𝐶 → ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
13		eqeq2 2835	. . . 4 ⊢ (𝑧 = 𝐶 → (𝐵 = 𝑧 ↔ 𝐵 = 𝐶))
14	12, 13	bibi12d 348	. . 3 ⊢ (𝑧 = 𝐶 → (((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
15	6, 10, 14	rspc3v 3638	. 2 ⊢ ((𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (∀𝑥 ∈ 𝑇 ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
16	2, 15	mpan9 509	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑇 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  (class class class)co 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161

This theorem is referenced by:  caovcand  7352  caofcan  40662