Theorem caov411 7360

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 7357	. . 3 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
7	6	oveq1i 7145	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷)
8		ovex 7168	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
9		caov.4	. . 3 ⊢ 𝐷 ∈ V
10	8, 3, 9, 5	caovass 7328	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷))
11		ovex 7168	. . 3 ⊢ (𝐶𝐹𝐵) ∈ V
12	11, 1, 9, 5	caovass 7328	. 2 ⊢ (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))
13	7, 10, 12	3eqtr3i 2829	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))

Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3441  (class class class)co 7135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138

This theorem is referenced by:  ecopovtrn  8383  distrnq  10372  lterpq  10381  ltanq  10382  prlem936  10458