Theorem caovcan 6248

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 6086	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2		oveq1 6086	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝐶) = (𝐴𝐹𝐶))
3	1, 2	eqeq12d 2253	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝐶)))
4	3	imbi1d 231	. 2 ⊢ (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶)))
5		oveq2 6087	. . . 4 ⊢ (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
6	5	eqeq1d 2247	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
7		eqeq1 2245	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = 𝐶 ↔ 𝐵 = 𝐶))
8	6, 7	imbi12d 234	. 2 ⊢ (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶)))
9		caovcan.1	. . 3 ⊢ 𝐶 ∈ V
10		oveq2 6087	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑥𝐹𝑧) = (𝑥𝐹𝐶))
11	10	eqeq2d 2250	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐹𝐶)))
12		eqeq2 2248	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝑦 = 𝑧 ↔ 𝑦 = 𝐶))
13	11, 12	imbi12d 234	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶)))
14	13	imbi2d 230	. . 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 2877	. 2 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶))
17	4, 8, 16	vtocl2ga 2891	1 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  (class class class)co 6079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6082

This theorem is referenced by: (None)