Theorem caov31 7587

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))

Assertion

Ref	Expression
caov31	⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)

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

Proof of Theorem caov31

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 ⊢ 𝐴 ∈ V
2		caov.3	. . . 4 ⊢ 𝐶 ∈ V
3		caov.2	. . . 4 ⊢ 𝐵 ∈ V
4		caov.ass	. . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
5	1, 2, 3, 4	caovass 7558	. . 3 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
6		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
7	1, 2, 3, 6, 4	caov12 7586	. . 3 ⊢ (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵))
8	5, 7	eqtri 2759	. 2 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
9	1, 3, 2, 6, 4	caov32 7585	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
10	2, 1, 3, 6, 4	caov32 7585	. . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴)
11	2, 1, 3, 4	caovass 7558	. . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
12	10, 11	eqtr3i 2761	. 2 ⊢ ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵))
13	8, 9, 12	3eqtr4i 2769	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440  (class class class)co 7358

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 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361

This theorem is referenced by:  caov13  7588  caov411  7590