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Theorem caov32d 5919
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
caov32d (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵))
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 caov32d
StepHypRef Expression
1 caovd.com . . . 4 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
2 caovd.2 . . . 4 (𝜑𝐵𝑆)
3 caovd.3 . . . 4 (𝜑𝐶𝑆)
41, 2, 3caovcomd 5895 . . 3 (𝜑 → (𝐵𝐹𝐶) = (𝐶𝐹𝐵))
54oveq2d 5758 . 2 (𝜑 → (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵)))
6 caovd.ass . . 3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7 caovd.1 . . 3 (𝜑𝐴𝑆)
86, 7, 2, 3caovassd 5898 . 2 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))
96, 7, 3, 2caovassd 5898 . 2 (𝜑 → ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵)))
105, 8, 93eqtr4d 2160 1 (𝜑 → ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947   = wceq 1316  wcel 1465  (class class class)co 5742
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-iota 5058  df-fv 5101  df-ov 5745
This theorem is referenced by:  caov31d  5921  mulcanenq  7161  mulcanenq0ec  7221  ltexprlemrl  7386  ltexprlemru  7388  cauappcvgprlemladdfl  7431  cauappcvgprlemladdru  7432  mulcmpblnrlemg  7516  ltsosr  7540  recexgt0sr  7549  mulgt0sr  7554  caucvgsrlemoffcau  7574  caucvgsrlemoffres  7576  resqrexlemover  10750
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