Theorem caopr32 4055

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
caopr32	|- ((AFB)FC) = ((AFC)FB)

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

Proof of Theorem caopr32

Step	Hyp	Ref	Expression
1		caopr.2	. . . 4 |- B e. V
2		caopr.3	. . . 4 |- C e. V
3		caopr.com	. . . 4 |- (xFy) = (yFx)
4	1, 2, 3	caoprcom 4048	. . 3 |- (BFC) = (CFB)
5	4	opreq2i 3967	. 2 |- (AF(BFC)) = (AF(CFB))
6		caopr.1	. . 3 |- A e. V
7		caopr.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
8	6, 1, 2, 7	caoprass 4049	. 2 |- ((AFB)FC) = (AF(BFC))
9	6, 2, 1, 7	caoprass 4049	. 2 |- ((AFC)FB) = (AF(CFB))
10	5, 8, 9	3eqtr4 1503	1 |- ((AFB)FC) = ((AFC)FB)

Syntax hints:   = wceq 955   e. wcel 957  Vcvv 1808  (class class class)co 3958

This theorem is referenced by:  caopr31 4057  distrpqlem 5049  ltexprlem7 5131  mulcmpblnrlem 5165  recexsrlem 5195  mulgt0sr 5197

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 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194  df-opr 3960

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