Theorem caov411 7382

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 7379	. . 3 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
7	6	oveq1i 7168	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷)
8		ovex 7191	. . 3 ⊢ (𝐴𝐹𝐵) ∈ V
9		caov.4	. . 3 ⊢ 𝐷 ∈ V
10	8, 3, 9, 5	caovass 7350	. 2 ⊢ (((𝐴𝐹𝐵)𝐹𝐶)𝐹𝐷) = ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷))
11		ovex 7191	. . 3 ⊢ (𝐶𝐹𝐵) ∈ V
12	11, 1, 9, 5	caovass 7350	. 2 ⊢ (((𝐶𝐹𝐵)𝐹𝐴)𝐹𝐷) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))
13	7, 10, 12	3eqtr3i 2854	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐶𝐹𝐵)𝐹(𝐴𝐹𝐷))

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3496  (class class class)co 7158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-nul 5212

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161

This theorem is referenced by:  ecopovtrn  8402  distrnq  10385  lterpq  10394  ltanq  10395  prlem936  10471