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Theorem caov13 7369
 Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caov.1 𝐴 ∈ V
caov.2 𝐵 ∈ V
caov.3 𝐶 ∈ V
caov.com (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caov.ass ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
Assertion
Ref Expression
caov13 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov13
StepHypRef Expression
1 caov.1 . . 3 𝐴 ∈ V
2 caov.2 . . 3 𝐵 ∈ V
3 caov.3 . . 3 𝐶 ∈ V
4 caov.com . . 3 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5 caov.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
61, 2, 3, 4, 5caov31 7368 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
71, 2, 3, 5caovass 7339 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
83, 2, 1, 5caovass 7339 . 2 ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐵𝐹𝐴))
96, 7, 83eqtr3i 2829 1 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3442  (class class class)co 7145 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3444  df-un 3888  df-in 3890  df-ss 3900  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-iota 6291  df-fv 6340  df-ov 7148 This theorem is referenced by:  ltsonq  10398  mulcmpblnrlem  10499
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