Theorem caovcan 7596

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

Hypotheses

Ref	Expression
caovcan.1	⊢ 𝐶 ∈ V
caovcan.2	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))

Assertion

Ref	Expression
caovcan	⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))

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

Proof of Theorem caovcan

Step	Hyp	Ref	Expression
1		oveq1 7399	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2		oveq1 7399	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝐶) = (𝐴𝐹𝐶))
3	1, 2	eqeq12d 2777	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝐶)))
4	3	imbi1d 343	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶)))
5		oveq2 7400	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
6	5	eqeq1d 2763	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
7		eqeq1 2765	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐶 ↔ 𝐵 = 𝐶))
8	6, 7	imbi12d 346	. 2 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶)))
9		caovcan.1	. . 3 ⊢ 𝐶 ∈ V
10		oveq2 7400	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑥𝐹𝑧) = (𝑥𝐹𝐶))
11	10	eqeq2d 2772	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐹𝐶)))
12		eqeq2 2773	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑦 = 𝑧 ↔ 𝑦 = 𝐶))
13	11, 12	imbi12d 346	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶)))
14	13	imbi2d 342	. . 3 ⊢ (𝑧 = 𝐶 → (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧)) ↔ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶))))
15		caovcan.2	. . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))
16	9, 14, 15	vtocl 3524	. 2 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶))
17	4, 8, 16	vtocl2ga 3542	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  (class class class)co 7392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-iota 6473  df-fv 6525  df-ov 7395

This theorem is referenced by:  ecopovtrn  8797