Theorem caopr411 4057

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
caopr411	|- ((AFB)F(CFD)) = ((CFB)F(AFD))

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 caopr411

Step	Hyp	Ref	Expression
1		caopr.1	. . . 4 |- A e. V
2		caopr.2	. . . 4 |- B e. V
3		caopr.3	. . . 4 |- C e. V
4		caopr.com	. . . 4 |- (xFy) = (yFx)
5		caopr.ass	. . . 4 |- ((xFy)Fz) = (xF(yFz))
6	1, 2, 3, 4, 5	caopr31 4054	. . 3 |- ((AFB)FC) = ((CFB)FA)
7	6	opreq1i 3962	. 2 |- (((AFB)FC)FD) = (((CFB)FA)FD)
8		oprex 3974	. . 3 |- (AFB) e. V
9		caopr.4	. . 3 |- D e. V
10	8, 3, 9, 5	caoprass 4046	. 2 |- (((AFB)FC)FD) = ((AFB)F(CFD))
11		oprex 3974	. . 3 |- (CFB) e. V
12	11, 1, 9, 5	caoprass 4046	. 2 |- (((CFB)FA)FD) = ((CFB)F(AFD))
13	7, 10, 12	3eqtr3 1500	1 |- ((AFB)F(CFD)) = ((CFB)F(AFD))

Syntax hints:   = wceq 954   e. wcel 956  Vcvv 1807  (class class class)co 3954

This theorem is referenced by:  ecopoprtrn 4301  ltapq 5056

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

Copyright terms: Public domain