Theorem caoprcom 4045

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 3959	. . . 4 |- (x = A -> (xFy) = (AFy))
4		opreq2 3960	. . . 4 |- (x = A -> (yFx) = (yFA))
5	3, 4	eqeq12d 1486	. . 3 |- (x = A -> ((xFy) = (yFx) <-> (AFy) = (yFA)))
6		opreq2 3960	. . . 4 |- (y = B -> (AFy) = (AFB))
7		opreq1 3959	. . . 4 |- (y = B -> (yFA) = (BFA))
8	6, 7	eqeq12d 1486	. . 3 |- (y = B -> ((AFy) = (yFA) <-> (AFB) = (BFA)))
9	5, 8	sylan9bb 539	. 2 |- ((x = A /\ y = B) -> ((xFy) = (yFx) <-> (AFB) = (BFA)))
10		caoprcom.3	. 2 |- (xFy) = (yFx)
11	1, 2, 9, 10	vtocl2 1839	1 |- (AFB) = (BFA)

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

This theorem is referenced by:  caoprord2 4049  caopr32 4052  caopr12 4053  caopr42 4058  caoprdistrr 4059  caoprmo 4062  ecopoprdm 4299  ecopoprsym 4300  genpcl 5091

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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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