Theorem caov12 7631

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 7600	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
5	4	oveq1i 7414	. 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 7603	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
9	2, 1, 6, 7	caovass 7603	. 2 ⊢ ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶))
10	5, 8, 9	3eqtr3i 2762	1 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3468  (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 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407

This theorem is referenced by:  caov31  7632  caov4  7634  caovmo  7640  distrnq  10955  ltaddnq  10968  ltexnq  10969  1idpr  11023  prlem934  11027  prlem936  11041  mulcmpblnrlem  11064  ltsosr  11088  0idsr  11091  1idsr  11092  recexsrlem  11097  mulgt0sr  11099  axmulass  11151