Theorem caopr12 3075

Description: Rearrange arguments in a commutative, associative operation.

Hypotheses

Ref	Expression
caopr.1	⊢ A ∈ V
caopr.2	⊢ B ∈ V
caopr.3	⊢ C ∈ V
caopr.com	⊢ (xFy) = (yFx)
caopr.ass	⊢ ((xFy)Fz) = (xF(yFz))

Assertion

Ref	Expression
caopr12	⊢ (AF(BFC)) = (BF(AFC))

Distinct variable group(s):   x,y,z,F   x,A,y,z   x,B,y,z   x,C,y,z

Proof of Theorem caopr12

Step	Hyp	Ref	Expression
1		caopr.1	. . . 4 ⊢ A ∈ V
2		caopr.2	. . . 4 ⊢ B ∈ V
3		caopr.com	. . . 4 ⊢ (xFy) = (yFx)
4	1, 2, 3	caoprcom 3067	. . 3 ⊢ (AFB) = (BFA)
5	4	opreq1i 3009	. 2 ⊢ ((AFB)FC) = ((BFA)FC)
6		caopr.3	. . 3 ⊢ C ∈ V
7		caopr.ass	. . 3 ⊢ ((xFy)Fz) = (xF(yFz))
8	1, 2, 6, 7	caoprass 3068	. 2 ⊢ ((AFB)FC) = (AF(BFC))
9	2, 1, 6, 7	caoprass 3068	. 2 ⊢ ((BFA)FC) = (BF(AFC))
10	5, 8, 9	3eqtr3 1124	1 ⊢ (AF(BFC)) = (BF(AFC))

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

This theorem is referenced by:  caopr31 3076  caopr4 3078  caoprmo 3084  ltsopq 3869  ltexpq 3874  1idpr 3927  prlem934b 3932  mulcmpblnrlem 3976  ltsosr 3997  0idsr 4000  1idsr 4001  recexsrlem 4006  mulgt0sr 4008  axmulass 4073  axrecex 4079

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-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-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.html">df-xp 2424  df-cnv 2426  df-dm 2428df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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