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Theorem caov32 7628
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
caov32 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov32
StepHypRef Expression
1 caov.2 . . . 4 𝐵 ∈ V
2 caov.3 . . . 4 𝐶 ∈ V
3 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
41, 2, 3caovcom 7598 . . 3 (𝐵𝐹𝐶) = (𝐶𝐹𝐵)
54oveq2i 7413 . 2 (𝐴𝐹(𝐵𝐹𝐶)) = (𝐴𝐹(𝐶𝐹𝐵))
6 caov.1 . . 3 𝐴 ∈ V
7 caov.ass . . 3 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
86, 1, 2, 7caovass 7601 . 2 ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
96, 2, 1, 7caovass 7601 . 2 ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
105, 8, 93eqtr4i 2762 1 ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  Vcvv 3466  (class class class)co 7402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405
This theorem is referenced by:  caov31  7630  addassnq  10950  ltexprlem7  11034  mulcmpblnrlem  11062  recexsrlem  11095  mulgt0sr  11097
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