Theorem ertr 5332

Description: An equivalence relation is transitive.

Hypotheses

Ref	Expression
ertr.1	|- A e. _V
ertr.2	|- B e. _V
ertr.3	|- C e. _V
ertr.4	|- Er R

Assertion

Ref	Expression
ertr	|- ((ARB /\ BRC) -> ARC)

Proof of Theorem ertr

Step	Hyp	Ref	Expression
1		ertr.1	. 2 |- A e. _V
2		ertr.2	. 2 |- B e. _V
3		ertr.3	. 2 |- C e. _V
4		breq1 3341	. . . . 5 |- (x = A -> (xRy <-> ARy))
5	4	anbi1d 679	. . . 4 |- (x = A -> ((xRy /\ yRz) <-> (ARy /\ yRz)))
6		breq1 3341	. . . 4 |- (x = A -> (xRz <-> ARz))
7	5, 6	imbi12d 688	. . 3 |- (x = A -> (((xRy /\ yRz) -> xRz) <-> ((ARy /\ yRz) -> ARz)))
8		breq2 3342	. . . . 5 |- (y = B -> (ARy <-> ARB))
9		breq1 3341	. . . . 5 |- (y = B -> (yRz <-> BRz))
10	8, 9	anbi12d 690	. . . 4 |- (y = B -> ((ARy /\ yRz) <-> (ARB /\ BRz)))
11	10	imbi1d 675	. . 3 |- (y = B -> (((ARy /\ yRz) -> ARz) <-> ((ARB /\ BRz) -> ARz)))
12		breq2 3342	. . . . 5 |- (z = C -> (BRz <-> BRC))
13	12	anbi2d 678	. . . 4 |- (z = C -> ((ARB /\ BRz) <-> (ARB /\ BRC)))
14		breq2 3342	. . . 4 |- (z = C -> (ARz <-> ARC))
15	13, 14	imbi12d 688	. . 3 |- (z = C -> (((ARB /\ BRz) -> ARz) <-> ((ARB /\ BRC) -> ARC)))
16	7, 11, 15	syl3an9b 1166	. 2 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
17		ertr.4	. . . . . . 7 |- Er R
18		dfer2 5319	. . . . . . 7 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
19	17, 18	mpbi 206	. . . . . 6 |- A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
20	19	a4i 1328	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
21	20	a4i 1328	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
22	21	a4i 1328	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
23	22	simpri 351	. 2 |- ((xRy /\ yRz) -> xRz)
24	1, 2, 3, 16, 23	vtocl3 2342	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292   class class class wbr 3338  Er wer 5315

This theorem is referenced by:  erref 5333  erthi 5339  erdisj 5344  entr 5473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-er 5318

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