Theorem caoprcom 4059

Description: Convert an operation commutative law to class notation.

Hypotheses

Ref	Expression
caoprcom.1	|- A e. V
caoprcom.2	|- B e. V
caoprcom.3	|- (xFy) = (yFx)

Assertion

Ref	Expression
caoprcom	|- (AFB) = (BFA)

Distinct variable groups:   x,y,F   x,A,y   x,B,y

Proof of Theorem caoprcom

Step	Hyp	Ref	Expression
1		caoprcom.1	. 2 |- A e. V
2		caoprcom.2	. 2 |- B e. V
3		opreq1 3974	. . . 4 |- (x = A -> (xFy) = (AFy))
4		opreq2 3975	. . . 4 |- (x = A -> (yFx) = (yFA))
5	3, 4	eqeq12d 1492	. . 3 |- (x = A -> ((xFy) = (yFx) <-> (AFy) = (yFA)))
6		opreq2 3975	. . . 4 |- (y = B -> (AFy) = (AFB))
7		opreq1 3974	. . . 4 |- (y = B -> (yFA) = (BFA))
8	6, 7	eqeq12d 1492	. . 3 |- (y = B -> ((AFy) = (yFA) <-> (AFB) = (BFA)))
9	5, 8	sylan9bb 542	. 2 |- ((x = A /\ y = B) -> ((xFy) = (yFx) <-> (AFB) = (BFA)))
10		caoprcom.3	. 2 |- (xFy) = (yFx)
11	1, 2, 9, 10	vtocl2 1846	1 |- (AFB) = (BFA)

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814  (class class class)co 3969

This theorem is referenced by:  caoprord2 4063  caopr32 4066  caopr12 4067  caopr42 4072  caoprdistrr 4073  caoprmo 4076  ecopoprdm 4315  ecopoprsym 4316  genpcl 5123

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204  df-opr 3971

Copyright terms: Public domain