Theorem caov12 7624

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
caov12	⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))

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

Proof of Theorem caov12

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 ⊢ 𝐴 ∈ V
2		caov.2	. . . 4 ⊢ 𝐵 ∈ V
3		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
4	1, 2, 3	caovcom 7593	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
5	4	oveq1i 7406	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶)
6		caov.3	. . 3 ⊢ 𝐶 ∈ V
7		caov.ass	. . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
8	1, 2, 6, 7	caovass 7596	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
9	2, 1, 6, 7	caovass 7596	. 2 ⊢ ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶))
10	5, 8, 9	3eqtr3i 2793	1 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))

Syntax hints:   = wceq 1560   ∈ wcel 2142  Vcvv 3454  (class class class)co 7396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-ov 7399

This theorem is referenced by:  caov31  7625  caov4  7627  caovmo  7633  distrnq  10919  ltaddnq  10932  ltexnq  10933  1idpr  10987  prlem934  10991  prlem936  11005  mulcmpblnrlem  11028  ltsosr  11052  0idsr  11055  1idsr  11056  recexsrlem  11061  mulgt0sr  11063  axmulass  11115