Theorem caov4 7651

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
caov4	⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))

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 caov4

Step	Hyp	Ref	Expression
1		caov.2	. . . 4 ⊢ 𝐵 ∈ V
2		caov.3	. . . 4 ⊢ 𝐶 ∈ V
3		caov.4	. . . 4 ⊢ 𝐷 ∈ V
4		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5		caov.ass	. . . 4 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
6	1, 2, 3, 4, 5	caov12 7648	. . 3 ⊢ (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))
7	6	oveq2i 7430	. 2 ⊢ (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
8		caov.1	. . 3 ⊢ 𝐴 ∈ V
9		ovex 7452	. . 3 ⊢ (𝐶𝐹𝐷) ∈ V
10	8, 1, 9, 5	caovass 7620	. 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))
11		ovex 7452	. . 3 ⊢ (𝐵𝐹𝐷) ∈ V
12	8, 2, 11, 5	caovass 7620	. 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
13	7, 10, 12	3eqtr4i 2798	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3457  (class class class)co 7419

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 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422

This theorem is used by:  caov42  7653  ecopovtrn  8824  adderpqlem  10954  mulerpqlem  10955  ltmnq  10972  reclem3pr  11049  mulcmpblnrlem  11070  distrsr  11091  ltasr  11100  mulgt0sr  11105  axdistr  11158