Theorem caovdir 7016

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

Step	Hyp	Ref	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))
5	1, 2, 3, 4	caovdi 7001	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6		ovex 6824	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
7		caovdir.com	. . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
8	1, 6, 7	caovcom 6979	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
9	1, 2, 7	caovcom 6979	. . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
10	1, 3, 7	caovcom 6979	. . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
11	9, 10	oveq12i 6806	. 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
12	5, 8, 11	3eqtr3i 2801	1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))

Syntax hints:   = wceq 1631   ∈ wcel 2145  Vcvv 3351  (class class class)co 6794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5995  df-fv 6040  df-ov 6797

This theorem is referenced by:  caovdilem  7017  adderpqlem  9979  addassnq  9983  prlem934  10058  prlem936  10072  recexsrlem  10127  mulgt0sr  10129