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Theorem caovcang 5972
 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 2537 . 2 (𝜑 → ∀𝑥𝑇𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧))
3 oveq1 5821 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦) = (𝐴𝐹𝑦))
4 oveq1 5821 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑧) = (𝐴𝐹𝑧))
53, 4eqeq12d 2169 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ (𝐴𝐹𝑦) = (𝐴𝐹𝑧)))
65bibi1d 232 . . 3 (𝑥 = 𝐴 → (((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧)))
7 oveq2 5822 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐹𝑦) = (𝐴𝐹𝐵))
87eqeq1d 2163 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝑧)))
9 eqeq1 2161 . . . 4 (𝑦 = 𝐵 → (𝑦 = 𝑧𝐵 = 𝑧))
108, 9bibi12d 234 . . 3 (𝑦 = 𝐵 → (((𝐴𝐹𝑦) = (𝐴𝐹𝑧) ↔ 𝑦 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧)))
11 oveq2 5822 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐹𝑧) = (𝐴𝐹𝐶))
1211eqeq2d 2166 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ (𝐴𝐹𝐵) = (𝐴𝐹𝐶)))
13 eqeq2 2164 . . . 4 (𝑧 = 𝐶 → (𝐵 = 𝑧𝐵 = 𝐶))
1412, 13bibi12d 234 . . 3 (𝑧 = 𝐶 → (((𝐴𝐹𝐵) = (𝐴𝐹𝑧) ↔ 𝐵 = 𝑧) ↔ ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
156, 10, 14rspc3v 2829 . 2 ((𝐴𝑇𝐵𝑆𝐶𝑆) → (∀𝑥𝑇𝑦𝑆𝑧𝑆 ((𝑥𝐹𝑦) = (𝑥𝐹𝑧) ↔ 𝑦 = 𝑧) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶)))
162, 15mpan9 279 1 ((𝜑 ∧ (𝐴𝑇𝐵𝑆𝐶𝑆)) → ((𝐴𝐹𝐵) = (𝐴𝐹𝐶) ↔ 𝐵 = 𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 2125  ∀wral 2432  (class class class)co 5814 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-v 2711  df-un 3102  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-iota 5128  df-fv 5171  df-ov 5817 This theorem is referenced by:  caovcand  5973
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