Theorem caov42 7483

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
caov42	⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))

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 caov42

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		caov.4	. . 3 ⊢ 𝐷 ∈ V
7	1, 2, 3, 4, 5, 6	caov4 7481	. 2 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷))
8	2, 6, 4	caovcom 7447	. . 3 ⊢ (𝐵𝐹𝐷) = (𝐷𝐹𝐵)
9	8	oveq2i 7266	. 2 ⊢ ((𝐴𝐹𝐶)𝐹(𝐵𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))
10	7, 9	eqtri 2766	1 ⊢ ((𝐴𝐹𝐵)𝐹(𝐶𝐹𝐷)) = ((𝐴𝐹𝐶)𝐹(𝐷𝐹𝐵))

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  caovlem2  7486  mulcmpblnrlem  10757  ltasr  10787  axmulass  10844