Theorem caoprcom 3067

Description: Convert an operation commutative law to class notation.

Hypotheses

Ref	Expression
caoprcom.1	⊢ A ∈ V
caoprcom.2	⊢ B ∈ V
caoprcom.3	⊢ (xFy) = (yFx)

Assertion

Ref	Expression
caoprcom	⊢ (AFB) = (BFA)

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

Proof of Theorem caoprcom

Step	Hyp	Ref	Expression
1		caoprcom.1	. 2 ⊢ A ∈ V
2		caoprcom.2	. 2 ⊢ B ∈ V
3		opreq1 3006	. . . 4 ⊢ (x = A → (xFy) = (AFy))
4		opreq2 3007	. . . 4 ⊢ (x = A → (yFx) = (yFA))
5	3, 4	cleq12d 1115	. . 3 ⊢ (x = A → ((xFy) = (yFx) ↔ (AFy) = (yFA)))
6		opreq2 3007	. . . 4 ⊢ (y = B → (AFy) = (AFB))
7		opreq1 3006	. . . 4 ⊢ (y = B → (yFA) = (BFA))
8	6, 7	cleq12d 1115	. . 3 ⊢ (y = B → ((AFy) = (yFA) ↔ (AFB) = (BFA)))
9	5, 8	sylan9bb 418	. 2 ⊢ ((x = A ∧ y = B) → ((xFy) = (yFx) ↔ (AFB) = (BFA)))
10		caoprcom.3	. 2 ⊢ (xFy) = (yFx)
11	1, 2, 9, 10	vtocl2 1379	1 ⊢ (AFB) = (BFA)

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

This theorem is referenced by:  caoprord2 3071  caopr32 3074  caopr12 3075  caopr42 3080  caoprdistrr 3081  caoprmo 3084  ecopoprdm 3245  ecopoprsym 3246  genpcl 3905

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-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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