Theorem ersym 4265

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 2620	. . 3 |- ((x = A /\ y = B) -> (xRy <-> ARB))
4		breq12 2620	. . . 4 |- ((y = B /\ x = A) -> (yRx <-> BRA))
5	4	ancoms 436	. . 3 |- ((x = A /\ y = B) -> (yRx <-> BRA))
6	3, 5	imbi12d 625	. 2 |- ((x = A /\ y = B) -> ((xRy -> yRx) <-> (ARB -> BRA)))
7		ersym.3	. . . . . . 7 |- Er R
8		dfer2 4255	. . . . . . 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 981	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
11	10	a4i 981	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
12	11	a4i 981	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
13	12	pm3.26i 320	. 2 |- (xRy -> yRx)
14	1, 2, 6, 13	vtocl2 1840	1 |- (ARB -> BRA)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  Vcvv 1808   class class class wbr 2615  Er wer 4251

This theorem is referenced by:  ersymb 4266  erth 4275  ensymg 4401

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-cnv 3182  df-co 3183  df-er 4254

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