Theorem caopr32 3074

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
caopr32	⊢ ((AFB)FC) = ((AFC)FB)

Distinct variable group(s):   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 ∈ V
2		caopr.3	. . . 4 ⊢ C ∈ V
3		caopr.com	. . . 4 ⊢ (xFy) = (yFx)
4	1, 2, 3	caoprcom 3067	. . 3 ⊢ (BFC) = (CFB)
5	4	opreq2i 3010	. 2 ⊢ (AF(BFC)) = (AF(CFB))
6		caopr.1	. . 3 ⊢ A ∈ V
7		caopr.ass	. . 3 ⊢ ((xFy)Fz) = (xF(yFz))
8	6, 1, 2, 7	caoprass 3068	. 2 ⊢ ((AFB)FC) = (AF(BFC))
9	6, 2, 1, 7	caoprass 3068	. 2 ⊢ ((AFC)FB) = (AF(CFB))
10	5, 8, 9	3eqtr4 1126	1 ⊢ ((AFB)FC) = ((AFC)FB)

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

This theorem is referenced by:  caopr31 3076  distrpqlem 3860  ltexprlem7 3942  mulcmpblnrlem 3976  recexsrlem 4006  mulgt0sr 4008

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 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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