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Theorem caov4 6741
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 6738 . . 3 (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))
76oveq2i 6538 . 2 (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
8 caov.1 . . 3 𝐴 ∈ V
9 ovex 6555 . . 3 (𝐶𝐹𝐷) ∈ V
108, 1, 9, 5caovass 6710 . 2 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))
11 ovex 6555 . . 3 (𝐵𝐹𝐷) ∈ V
128, 2, 11, 5caovass 6710 . 2 ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
137, 10, 123eqtr4i 2641 1 ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wcel 1976  Vcvv 3172  (class class class)co 6527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-iota 5754  df-fv 5798  df-ov 6530
This theorem is referenced by:  caov42  6743  ecopovtrn  7715  adderpqlem  9633  mulerpqlem  9634  ltmnq  9651  reclem3pr  9728  mulcmpblnrlem  9748  distrsr  9769  ltasr  9778  mulgt0sr  9783  axdistr  9836
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