Theorem caovdir 7580

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 7565	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6		ovex 7379	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
7		caovdir.com	. . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
8	1, 6, 7	caovcom 7543	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
9	1, 2, 7	caovcom 7543	. . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
10	1, 3, 7	caovcom 7543	. . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
11	9, 10	oveq12i 7358	. 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
12	5, 8, 11	3eqtr3i 2762	1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436  (class class class)co 7346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349

This theorem is referenced by:  caovdilem  7581  adderpqlem  10845  addassnq  10849  prlem934  10924  prlem936  10938  recexsrlem  10994  mulgt0sr  10996