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Theorem caov42 7365
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))
caov.4 𝐷 ∈ V
Assertion
Ref Expression
caov42 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov42
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))
6 caov.4 . . 3 𝐷 ∈ V
71, 2, 3, 4, 5, 6caov4 7363 . 2 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
82, 6, 4caovcom 7329 . . 3 (𝐵𝐹𝐷) = (𝐷𝐹𝐵)
98oveq2i 7150 . 2 ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))
107, 9eqtri 2824 1 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  Vcvv 3444  (class class class)co 7139
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-nul 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-iota 6287  df-fv 6336  df-ov 7142
This theorem is referenced by:  caovlem2  7368  mulcmpblnrlem  10485  ltasr  10515  axmulass  10572
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