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Theorem caovdir 6821
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 6806 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6 ovex 6632 . . 3 (𝐴𝐹𝐵) ∈ V
7 caovdir.com . . 3 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
81, 6, 7caovcom 6784 . 2 (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
91, 2, 7caovcom 6784 . . 3 (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
101, 3, 7caovcom 6784 . . 3 (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
119, 10oveq12i 6616 . 2 ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
125, 8, 113eqtr3i 2651 1 ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  Vcvv 3186  (class class class)co 6604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607
This theorem is referenced by:  caovdilem  6822  adderpqlem  9720  addassnq  9724  prlem934  9799  prlem936  9813  recexsrlem  9868  mulgt0sr  9870
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