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Theorem caovdir 7668
Description: Reverse distributive law. (Contributed by NM, 26-Aug-1995.)
Hypotheses
Ref Expression
caovdir.1 𝐴 ∈ V
caovdir.2 𝐵 ∈ V
caovdir.3 𝐶 ∈ V
caovdir.com (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
caovdir.distr (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
Assertion
Ref Expression
caovdir ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem caovdir
StepHypRef Expression
1 caovdir.3 . . 3 𝐶 ∈ V
2 caovdir.1 . . 3 𝐴 ∈ V
3 caovdir.2 . . 3 𝐵 ∈ V
4 caovdir.distr . . 3 (𝑥𝐺(𝑦𝐹𝑧)) = ((𝑥𝐺𝑦)𝐹(𝑥𝐺𝑧))
51, 2, 3, 4caovdi 7653 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6 ovex 7465 . . 3 (𝐴𝐹𝐵) ∈ V
7 caovdir.com . . 3 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
81, 6, 7caovcom 7631 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
91, 2, 7caovcom 7631 . . 3 (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
101, 3, 7caovcom 7631 . . 3 (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
119, 10oveq12i 7444 . 2 ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
125, 8, 113eqtr3i 2772 1 ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  wcel 2107  Vcvv 3479  (class class class)co 7432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-nul 5305
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  caovdilem  7669  adderpqlem  10995  addassnq  10999  prlem934  11074  prlem936  11088  recexsrlem  11144  mulgt0sr  11146
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