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Theorem caov31d 5734
Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.) (Revised by Mario Carneiro, 30-Dec-2014.)
Hypotheses
Ref Expression
caovd.1 (𝜑𝐴𝑆)
caovd.2 (𝜑𝐵𝑆)
caovd.3 (𝜑𝐶𝑆)
caovd.com ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovd.ass ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
Assertion
Ref Expression
caov31d (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴))
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

Proof of Theorem caov31d
StepHypRef Expression
1 caovd.com . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2 caovd.1 . . . 4 (𝜑𝐴𝑆)
3 caovd.3 . . . 4 (𝜑𝐶𝑆)
41, 2, 3caovcomd 5708 . . 3 (𝜑 → (𝐴𝐹𝐶) = (𝐶𝐹𝐴))
54oveq1d 5578 . 2 (𝜑 → ((𝐴𝐹𝐶)𝐹𝐵) = ((𝐶𝐹𝐴)𝐹𝐵))
6 caovd.2 . . 3 (𝜑𝐵𝑆)
7 caovd.ass . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
82, 6, 3, 1, 7caov32d 5732 . 2 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵))
93, 6, 2, 1, 7caov32d 5732 . 2 (𝜑 → ((𝐶𝐹𝐵)𝐹𝐴) = ((𝐶𝐹𝐴)𝐹𝐵))
105, 8, 93eqtr4d 2125 1 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5563
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-iota 4917  df-fv 4960  df-ov 5566
This theorem is referenced by:  caov13d  5735
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