Theorem caovdir 7636

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 7621	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵))
6		ovex 7437	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
7		caovdir.com	. . 3 ⊢ (𝑥𝐺𝑦) = (𝑦𝐺𝑥)
8	1, 6, 7	caovcom 7599	. 2 ⊢ (𝐶𝐺(𝐴𝐹𝐵)) = ((𝐴𝐹𝐵)𝐺𝐶)
9	1, 2, 7	caovcom 7599	. . 3 ⊢ (𝐶𝐺𝐴) = (𝐴𝐺𝐶)
10	1, 3, 7	caovcom 7599	. . 3 ⊢ (𝐶𝐺𝐵) = (𝐵𝐺𝐶)
11	9, 10	oveq12i 7416	. 2 ⊢ ((𝐶𝐺𝐴)𝐹(𝐶𝐺𝐵)) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))
12	5, 8, 11	3eqtr3i 2769	1 ⊢ ((𝐴𝐹𝐵)𝐺𝐶) = ((𝐴𝐺𝐶)𝐹(𝐵𝐺𝐶))

Syntax hints:   = wceq 1542   ∈ wcel 2107  Vcvv 3475  (class class class)co 7404

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407

This theorem is referenced by:  caovdilem  7637  adderpqlem  10945  addassnq  10949  prlem934  11024  prlem936  11038  recexsrlem  11094  mulgt0sr  11096