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Theorem caov31 6011
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 5982 . . 3 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
71, 2, 3, 6, 4caov12 6010 . . 3 (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵))
85, 7eqtri 2178 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
91, 3, 2, 6, 4caov32 6009 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
102, 1, 3, 6, 4caov32 6009 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴)
112, 1, 3, 4caovass 5982 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
1210, 11eqtr3i 2180 . 2 ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵))
138, 9, 123eqtr4i 2188 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1335  wcel 2128  Vcvv 2712  (class class class)co 5825
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-iota 5136  df-fv 5179  df-ov 5828
This theorem is referenced by:  caov13  6012
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