Theorem caopr12 4067

Description: Rearrange arguments in a commutative, associative operation.

Hypotheses

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

Assertion

Ref	Expression
caopr12	|- (AF(BFC)) = (BF(AFC))

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

Proof of Theorem caopr12

Step	Hyp	Ref	Expression
1		caopr.1	. . . 4 |- A e. V
2		caopr.2	. . . 4 |- B e. V
3		caopr.com	. . . 4 |- (xFy) = (yFx)
4	1, 2, 3	caoprcom 4059	. . 3 |- (AFB) = (BFA)
5	4	opreq1i 3977	. 2 |- ((AFB)FC) = ((BFA)FC)
6		caopr.3	. . 3 |- C e. V
7		caopr.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
8	1, 2, 6, 7	caoprass 4060	. 2 |- ((AFB)FC) = (AF(BFC))
9	2, 1, 6, 7	caoprass 4060	. 2 |- ((BFA)FC) = (BF(AFC))
10	5, 8, 9	3eqtr3 1506	1 |- (AF(BFC)) = (BF(AFC))

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  (class class class)co 3969

This theorem is referenced by:  caopr31 4068  caopr4 4070  caoprmo 4076  ltsopq 5087  ltexpq 5092  1idpr 5145  prlem934b 5150  mulcmpblnrlem 5194  ltsosr 5215  0idsr 5218  1idsr 5219  recexsrlem 5224  mulgt0sr 5226  axmulass 5290

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

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