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Theorem caov411d 5764
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)))
caovd.4 (𝜑𝐷𝑆)
caovd.cl ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
Assertion
Ref Expression
caov411d (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem caov411d
StepHypRef Expression
1 caovd.2 . . 3 (𝜑𝐵𝑆)
2 caovd.1 . . 3 (𝜑𝐴𝑆)
3 caovd.3 . . 3 (𝜑𝐶𝑆)
4 caovd.com . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
5 caovd.ass . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
6 caovd.4 . . 3 (𝜑𝐷𝑆)
7 caovd.cl . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
81, 2, 3, 4, 5, 6, 7caov4d 5763 . 2 (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷)))
94, 1, 2caovcomd 5735 . . 3 (𝜑 → (𝐵𝐹𝐴) = (𝐴𝐹𝐵))
109oveq1d 5605 . 2 (𝜑 → ((𝐵𝐹𝐴)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)))
114, 1, 3caovcomd 5735 . . 3 (𝜑 → (𝐵𝐹𝐶) = (𝐶𝐹𝐵))
1211oveq1d 5605 . 2 (𝜑 → ((𝐵𝐹𝐶)𝐹(𝐴𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))
138, 10, 123eqtr3d 2123 1 (𝜑 → ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920   = wceq 1285  wcel 1434  (class class class)co 5590
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 2614  df-un 2988  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-iota 4933  df-fv 4976  df-ov 5593
This theorem is referenced by:  ecopovtrn  6318  ecopovtrng  6321  ltsonq  6859  ltanqg  6861  mulextsr1lem  7227
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