Theorem caovdir 7659

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 7644	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6		ovex 7457	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
7		caovdir.com	. . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
8	1, 6, 7	caovcom 7622	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
9	1, 2, 7	caovcom 7622	. . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
10	1, 3, 7	caovcom 7622	. . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
11	9, 10	oveq12i 7436	. 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
12	5, 8, 11	3eqtr3i 2763	1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3471  (class class class)co 7424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698  ax-nul 5308

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-iota 6503  df-fv 6559  df-ov 7427

This theorem is referenced by:  caovdilem  7660  adderpqlem  10983  addassnq  10987  prlem934  11062  prlem936  11076  recexsrlem  11132  mulgt0sr  11134