Theorem caov4 7638

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 7635	. . 3 ⊢ (𝐵𝐹(𝐶𝐹𝐷)) = (𝐶𝐹(𝐵𝐹𝐷))
7	6	oveq2i 7416	. 2 ⊢ (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷))) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
8		caov.1	. . 3 ⊢ 𝐴 ∈ V
9		ovex 7438	. . 3 ⊢ (𝐶𝐹𝐷) ∈ V
10	8, 1, 9, 5	caovass 7607	. 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = (𝐴𝐹(𝐵𝐹(𝐶𝐹𝐷)))
11		ovex 7438	. . 3 ⊢ (𝐵𝐹𝐷) ∈ V
12	8, 2, 11, 5	caovass 7607	. 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = (𝐴𝐹(𝐶𝐹(𝐵𝐹𝐷)))
13	7, 10, 12	3eqtr4i 2768	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3459  (class class class)co 7405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by:  caov42  7640  ecopovtrn  8834  adderpqlem  10968  mulerpqlem  10969  ltmnq  10986  reclem3pr  11063  mulcmpblnrlem  11084  distrsr  11105  ltasr  11114  mulgt0sr  11119  axdistr  11172