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Theorem caov31 7236
 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 7207 . . 3 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
71, 2, 3, 6, 4caov12 7235 . . 3 (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵))
85, 7eqtri 2818 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
91, 3, 2, 6, 4caov32 7234 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
102, 1, 3, 6, 4caov32 7234 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴)
112, 1, 3, 4caovass 7207 . . 3 ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
1210, 11eqtr3i 2820 . 2 ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵))
138, 9, 123eqtr4i 2828 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∈ wcel 2080  Vcvv 3436  (class class class)co 7019 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-ext 2768 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-iota 6192  df-fv 6236  df-ov 7022 This theorem is referenced by:  caov13  7237  caov411  7239
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