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Theorem caovcan 7005
Description: Convert an operation cancellation law to class notation. (Contributed by NM, 20-Aug-1995.)
Hypotheses
Ref Expression
caovcan.1 𝐶 ∈ V
caovcan.2 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))
Assertion
Ref Expression
caovcan ((𝐴𝑆𝐵𝑆) → ((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
StepHypRef Expression
1 oveq1 6822 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
2 oveq1 6822 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹𝐶) = (𝐴𝐹𝐶))
31, 2eqeq12d 2776 . . 3 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝐶)))
43imbi1d 330 . 2 (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶)))
5 oveq2 6823 . . . 4 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
65eqeq1d 2763 . . 3 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝐶) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
7 eqeq1 2765 . . 3 (𝑦 = 𝐵 → (𝑦 = 𝐶𝐵 = 𝐶))
86, 7imbi12d 333 . 2 (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝐶) → 𝑦 = 𝐶) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶)))
9 caovcan.1 . . 3 𝐶 ∈ V
10 oveq2 6823 . . . . . 6 (𝑧 = 𝐶 → (𝑥𝐹𝑧) = (𝑥𝐹𝐶))
1110eqeq2d 2771 . . . . 5 (𝑧 = 𝐶 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝑥𝐹𝑦) = (𝑥𝐹𝐶)))
12 eqeq2 2772 . . . . 5 (𝑧 = 𝐶 → (𝑦 = 𝑧𝑦 = 𝐶))
1311, 12imbi12d 333 . . . 4 (𝑧 = 𝐶 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶)))
1413imbi2d 329 . . 3 (𝑧 = 𝐶 → (((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧)) ↔ ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶))))
15 caovcan.2 . . 3 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) → 𝑦 = 𝑧))
169, 14, 15vtocl 3400 . 2 ((𝑥𝑆𝑦𝑆) → ((𝑥𝐹𝑦) = (𝑥𝐹𝐶) → 𝑦 = 𝐶))
174, 8, 16vtocl2ga 3415 1 ((𝐴𝑆𝐵𝑆) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) → 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  Vcvv 3341  (class class class)co 6815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-iota 6013  df-fv 6058  df-ov 6818
This theorem is referenced by:  ecopovtrn  8020
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