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Theorem caov12 7194
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
caov12 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov12
StepHypRef Expression
1 caov.1 . . . 4 𝐴 ∈ V
2 caov.2 . . . 4 𝐵 ∈ V
3 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
41, 2, 3caovcom 7163 . . 3 (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
54oveq1i 6988 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶)
6 caov.3 . . 3 𝐶 ∈ V
7 caov.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
81, 2, 6, 7caovass 7166 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
92, 1, 6, 7caovass 7166 . 2 ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶))
105, 8, 93eqtr3i 2810 1 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1507  wcel 2050  Vcvv 3415  (class class class)co 6978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-iota 6154  df-fv 6198  df-ov 6981
This theorem is referenced by:  caov31  7195  caov4  7197  caovmo  7203  distrnq  10183  ltaddnq  10196  ltexnq  10197  1idpr  10251  prlem934  10255  prlem936  10269  mulcmpblnrlem  10292  ltsosr  10316  0idsr  10319  1idsr  10320  recexsrlem  10325  mulgt0sr  10327  axmulass  10379
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