Theorem caov13 6280

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
caov13	⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))

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

Proof of Theorem caov13

Step	Hyp	Ref	Expression
1		caov.1	. . 3 ⊢ 𝐴 ∈ V
2		caov.2	. . 3 ⊢ 𝐵 ∈ V
3		caov.3	. . 3 ⊢ 𝐶 ∈ V
4		caov.com	. . 3 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥)
5		caov.ass	. . 3 ⊢ ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧))
6	1, 2, 3, 4, 5	caov31 6279	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = ((𝐶𝐹𝐵)𝐹𝐴)
7	1, 2, 3, 5	caovass 6250	. 2 ⊢ ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))
8	3, 2, 1, 5	caovass 6250	. 2 ⊢ ((𝐶𝐹𝐵)𝐹𝐴) = (𝐶𝐹(𝐵𝐹𝐴))
9	6, 7, 8	3eqtr3i 2267	1 ⊢ (𝐴𝐹(𝐵𝐹𝐶)) = (𝐶𝐹(𝐵𝐹𝐴))

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by: (None)