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Theorem caov4 7646
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
caov4 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
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 caov4
StepHypRef Expression
1 caov.2 . . . 4 𝐵 ∈ V
2 caov.3 . . . 4 𝐶 ∈ V
3 caov.4 . . . 4 𝐷 ∈ V
4 caov.com . . . 4 (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5 caov.ass . . . 4 ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
61, 2, 3, 4, 5caov12 7643 . . 3 (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))
76oveq2i 7425 . 2 (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
8 caov.1 . . 3 𝐴 ∈ V
9 ovex 7447 . . 3 (𝐶𝐹𝐷) ∈ V
108, 1, 9, 5caovass 7615 . 2 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))
11 ovex 7447 . . 3 (𝐵𝐹𝐷) ∈ V
128, 2, 11, 5caovass 7615 . 2 ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
137, 10, 123eqtr4i 2765 1 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  Vcvv 3469  (class class class)co 7414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-nul 5300
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-iota 6494  df-fv 6550  df-ov 7417
This theorem is referenced by:  caov42  7648  ecopovtrn  8832  adderpqlem  10971  mulerpqlem  10972  ltmnq  10989  reclem3pr  11066  mulcmpblnrlem  11087  distrsr  11108  ltasr  11117  mulgt0sr  11122  axdistr  11175
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