Theorem caov31 7662

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 7633	. . 3 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐴𝐹(𝐶𝐹𝐵))
6		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
7	1, 2, 3, 6, 4	caov12 7661	. . 3 ⊢ (𝐴𝐹(𝐶𝐹𝐵)) = (𝐶𝐹(𝐴𝐹𝐵))
8	5, 7	eqtri 2765	. 2 ⊢ ((𝐴𝐹𝐶)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
9	1, 3, 2, 6, 4	caov32 7660	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐴𝐹𝐶)𝐹𝐵)
10	2, 1, 3, 6, 4	caov32 7660	. . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = ((𝐶𝐹𝐵)𝐹𝐴)
11	2, 1, 3, 4	caovass 7633	. . 3 ⊢ ((𝐶𝐹𝐴)𝐹𝐵) = (𝐶𝐹(𝐴𝐹𝐵))
12	10, 11	eqtr3i 2767	. 2 ⊢ ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐴𝐹𝐵))
13	8, 9, 12	3eqtr4i 2775	1 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  (class class class)co 7431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434

This theorem is referenced by:  caov13  7663  caov411  7665