Theorem caopr42 3080

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 group(s):   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 3078	. 2 |- ((AFB)F(CFD)) = ((AFC)F(BFD))
8	2, 6, 4	caoprcom 3067	. . 3 |- (BFD) = (DFB)
9	8	opreq2i 3010	. 2 |- ((AFC)F(BFD)) = ((AFC)F(DFB))
10	7, 9	eqtr 1119	1 |- ((AFB)F(CFD)) = ((AFC)F(DFB))

Syntax hints:   = wceq 1091   e. wcel 1092  Vcvv 1348  (class class class)co 3001

This theorem is referenced by:  caoprlem2 3083  prlem936 3949  mulcmpblnrlem 3976  ltasr 4003  axmulass 4073

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

metamath.org