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Theorem caovcang 6733
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 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
Assertion
Ref Expression
caovcang ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((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
StepHypRef Expression
1 caovcang.1 . . 3 ((𝜑 ∧ (𝑥𝑇𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
21ralrimivvva 2955 . 2 (𝜑 → ∀𝑥𝑇𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
3 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4 oveq1 6556 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑧) = (𝐴𝐹𝑧))
53, 4eqeq12d 2625 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝑧)))
65bibi1d 332 . . 3 (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧)))
7 oveq2 6557 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
87eqeq1d 2612 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝑧)))
9 eqeq1 2614 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
108, 9bibi12d 334 . . 3 (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧)))
11 oveq2 6557 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐹𝑧) = (𝐴𝐹𝐶))
1211eqeq2d 2620 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
13 eqeq2 2621 . . . 4 (𝑧 = 𝐶 → (𝐵 = 𝑧𝐵 = 𝐶))
1412, 13bibi12d 334 . . 3 (𝑧 = 𝐶 → (((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
156, 10, 14rspc3v 3296 . 2 ((𝐴𝑇𝐵𝑆𝐶𝑆) → (∀𝑥𝑇𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
162, 15mpan9 485 1 ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  (class class class)co 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552
This theorem is referenced by:  caovcand  6734  caofcan  37544
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