Theorem caov31 5638

Description: Rearrange arguments in a commutative, associative operation. (Contributed by set.mm contributors, 26-Aug-1995.)

Hypotheses

Ref	Expression
caopr.1	⊢ A ∈ V
caopr.2	⊢ B ∈ V
caopr.3	⊢ C ∈ V
caopr.com	⊢ (xFy) = (yFx)
caopr.ass	⊢ ((xFy)Fz) = (xF(yFz))

Assertion

Ref	Expression
caov31	⊢ ((AFB)FC) = ((CFB)FA)

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

Proof of Theorem caov31

Step	Hyp	Ref	Expression
1		caopr.1	. . . 4 ⊢ A ∈ V
2		caopr.3	. . . 4 ⊢ C ∈ V
3		caopr.2	. . . 4 ⊢ B ∈ V
4		caopr.ass	. . . 4 ⊢ ((xFy)Fz) = (xF(yFz))
5	1, 2, 3, 4	caovass 5628	. . 3 ⊢ ((AFC)FB) = (AF(CFB))
6		caopr.com	. . . 4 ⊢ (xFy) = (yFx)
7	1, 2, 3, 6, 4	caov12 5637	. . 3 ⊢ (AF(CFB)) = (CF(AFB))
8	5, 7	eqtri 2373	. 2 ⊢ ((AFC)FB) = (CF(AFB))
9	1, 3, 2, 6, 4	caov32 5636	. 2 ⊢ ((AFB)FC) = ((AFC)FB)
10	2, 1, 3, 6, 4	caov32 5636	. . 3 ⊢ ((CFA)FB) = ((CFB)FA)
11	2, 1, 3, 4	caovass 5628	. . 3 ⊢ ((CFA)FB) = (CF(AFB))
12	10, 11	eqtr3i 2375	. 2 ⊢ ((CFB)FA) = (CF(AFB))
13	8, 9, 12	3eqtr4i 2383	1 ⊢ ((AFB)FC) = ((CFB)FA)

Syntax hints:   = wceq 1642   ∈ wcel 1710  Vcvv 2860  (class class class)co 5526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796  df-ov 5527

This theorem is referenced by:  caov13  5639  caov411  5641