Theorem caov411 7672

Description: Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)

Hypotheses

Ref	Expression
caov.1	⊢ 𝐴 ∈ V
caov.2	⊢ 𝐵 ∈ V
caov.3	⊢ 𝐶 ∈ V
caov.com	⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
caov.ass	⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
caov.4	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
caov411	⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐷,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov411

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 ⊢ 𝐴 ∈ V
2		caov.2	. . . 4 ⊢ 𝐵 ∈ V
3		caov.3	. . . 4 ⊢ 𝐶 ∈ V
4		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5		caov.ass	. . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
6	1, 2, 3, 4, 5	caov31 7669	. . 3 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
7	6	oveq1i 7448	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷)
8		ovex 7471	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
9		caov.4	. . 3 ⊢ 𝐷 ∈ V
10	8, 3, 9, 5	caovass 7640	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷))
11		ovex 7471	. . 3 ⊢ (𝐶𝐹𝐵) ∈ V
12	11, 1, 9, 5	caovass 7640	. 2 ⊢ (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))
13	7, 10, 12	3eqtr3i 2773	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3481  (class class class)co 7438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5315

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577  df-ov 7441

This theorem is referenced by:  ecopovtrn  8868  distrnq  11008  lterpq  11017  ltanq  11018  prlem936  11094