Theorem ersym 4256

Description: An equivalence relation is symmetric.

Hypotheses

Ref	Expression
ersym.1	|- A e. V
ersym.2	|- B e. V
ersym.3	|- Er R

Assertion

Ref	Expression
ersym	|- (ARB -> BRA)

Proof of Theorem ersym

Step	Hyp	Ref	Expression
1		ersym.1	. 2 |- A e. V
2		ersym.2	. 2 |- B e. V
3		breq12 2614	. . 3 |- ((x = A /\ y = B) -> (xRy <-> ARB))
4		breq12 2614	. . . 4 |- ((y = B /\ x = A) -> (yRx <-> BRA))
5	4	ancoms 436	. . 3 |- ((x = A /\ y = B) -> (yRx <-> BRA))
6	3, 5	imbi12d 624	. 2 |- ((x = A /\ y = B) -> ((xRy -> yRx) <-> (ARB -> BRA)))
7		ersym.3	. . . . . . 7 |- Er R
8		dfer2 4246	. . . . . . 7 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
9	7, 8	mpbi 189	. . . . . 6 |- A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
10	9	a4i 979	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
11	10	a4i 979	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
12	11	a4i 979	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
13	12	pm3.26i 320	. 2 |- (xRy -> yRx)
14	1, 2, 6, 13	vtocl2 1834	1 |- (ARB -> BRA)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  Vcvv 1802   class class class wbr 2609  Er wer 4242

This theorem is referenced by:  ersymb 4257  erth 4266  ensymg 4392

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-cnv 3176  df-co 3177  df-er 4245

Copyright terms: Public domain