Theorem caopr31 4059

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
caopr31	|- ((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 caopr31

Step	Hyp	Ref	Expression
1		caopr.1	. . . 4 |- A e. V
2		caopr.3	. . . 4 |- C e. V
3		caopr.2	. . . 4 |- B e. V
4		caopr.ass	. . . 4 |- ((xFy)Fz) = (xF(yFz))
5	1, 2, 3, 4	caoprass 4051	. . 3 |- ((AFC)FB) = (AF(CFB))
6		caopr.com	. . . 4 |- (xFy) = (yFx)
7	1, 2, 3, 6, 4	caopr12 4058	. . 3 |- (AF(CFB)) = (CF(AFB))
8	5, 7	eqtr 1494	. 2 |- ((AFC)FB) = (CF(AFB))
9	1, 3, 2, 6, 4	caopr32 4057	. 2 |- ((AFB)FC) = ((AFC)FB)
10	2, 1, 3, 6, 4	caopr32 4057	. . 3 |- ((CFA)FB) = ((CFB)FA)
11	2, 1, 3, 4	caoprass 4051	. . 3 |- ((CFA)FB) = (CF(AFB))
12	10, 11	eqtr3 1496	. 2 |- ((CFB)FA) = (CF(AFB))
13	8, 9, 12	3eqtr4 1504	1 |- ((AFB)FC) = ((CFB)FA)

Syntax hints:   = wceq 955   e. wcel 957  Vcvv 1809  (class class class)co 3960

This theorem is referenced by:  caopr13 4060  caopr411 4062  prlem934b 5125  prlem936a 5140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-xp 3181  df-cnv 3183  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fv 3195  df-opr 3962

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