Theorem caov12 7656

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))

Assertion

Ref	Expression
caov12	⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧

Proof of Theorem caov12

Step	Hyp	Ref	Expression
1		caov.1	. . . 4 ⊢ 𝐴 ∈ V
2		caov.2	. . . 4 ⊢ 𝐵 ∈ V
3		caov.com	. . . 4 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
4	1, 2, 3	caovcom 7625	. . 3 ⊢ (𝐴𝐹𝐵) = (𝐵𝐹𝐴)
5	4	oveq1i 7437	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐵𝐹𝐴)𝐹𝐶)
6		caov.3	. . 3 ⊢ 𝐶 ∈ V
7		caov.ass	. . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
8	1, 2, 6, 7	caovass 7628	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
9	2, 1, 6, 7	caovass 7628	. 2 ⊢ ((𝐵𝐹𝐴)𝐹𝐶) = (𝐵𝐹(𝐴𝐹𝐶))
10	5, 8, 9	3eqtr3i 2794	1 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐵𝐹(𝐴𝐹𝐶))

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146  Vcvv 3453  (class class class)co 7427

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-iota 6500  df-fv 6554  df-ov 7430

This theorem is used by:  caov31  7657  caov4  7659  caovmo  7665  distrnq  11056  ltaddnq  11069  ltexnq  11070  1idpr  11124  prlem934  11128  prlem936  11142  mulcmpblnrlem  11165  ltsosr  11189  0idsr  11192  1idsr  11193  recexsrlem  11198  mulgt0sr  11200  axmulass  11252