Theorem caopr42 4066

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))
caopr.4	|- D e. V

Assertion

Ref	Expression
caopr42	|- ((AFB)F(CFD)) = ((AFC)F(DFB))

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

Proof of Theorem caopr42

Step	Hyp	Ref	Expression
1		caopr.1	. . 3 |- A e. V
2		caopr.2	. . 3 |- B e. V
3		caopr.3	. . 3 |- C e. V
4		caopr.com	. . 3 |- (xFy) = (yFx)
5		caopr.ass	. . 3 |- ((xFy)Fz) = (xF(yFz))
6		caopr.4	. . 3 |- D e. V
7	1, 2, 3, 4, 5, 6	caopr4 4064	. 2 |- ((AFB)F(CFD)) = ((AFC)F(BFD))
8	2, 6, 4	caoprcom 4053	. . 3 |- (BFD) = (DFB)
9	8	opreq2i 3972	. 2 |- ((AFC)F(BFD)) = ((AFC)F(DFB))
10	7, 9	eqtr 1495	1 |- ((AFB)F(CFD)) = ((AFC)F(DFB))

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811  (class class class)co 3963

This theorem is referenced by:  caoprlem2 4069  prlem936 5155  mulcmpblnrlem 5182  ltasr 5209  axmulass 5278

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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