Theorem caov411 7647

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 7644	. . 3 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
7	6	oveq1i 7424	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷)
8		ovex 7447	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
9		caov.4	. . 3 ⊢ 𝐷 ∈ V
10	8, 3, 9, 5	caovass 7615	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷))
11		ovex 7447	. . 3 ⊢ (𝐶𝐹𝐵) ∈ V
12	11, 1, 9, 5	caovass 7615	. 2 ⊢ (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))
13	7, 10, 12	3eqtr3i 2791	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  Vcvv 3450  (class class class)co 7414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417

This theorem is used by:  ecopovtrn  8823  distrnq  10973  lterpq  10982  ltanq  10983  prlem936  11059