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Theorem caov31 6860
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
caov31 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov31
StepHypRef Expression
1 caov.1 . . . 4 𝐴 ∈ V
2 caov.3 . . . 4 𝐶 ∈ V
3 caov.2 . . . 4 𝐵 ∈ V
4 caov.ass . . . 4 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
51, 2, 3, 4caovass 6831 . . 3 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
71, 2, 3, 6, 4caov12 6859 . . 3 (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵))
85, 7eqtri 2643 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
91, 3, 2, 6, 4caov32 6858 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
102, 1, 3, 6, 4caov32 6858 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴)
112, 1, 3, 4caovass 6831 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
1210, 11eqtr3i 2645 . 2 ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵))
138, 9, 123eqtr4i 2653 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  wcel 1989  Vcvv 3198  (class class class)co 6647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650
This theorem is referenced by:  caov13  6861  caov411  6863
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