Theorem ertr 4358

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 2672	. . . . 5 |- (x = A -> (xRy <-> ARy))
5	4	anbi1d 619	. . . 4 |- (x = A -> ((xRy /\ yRz) <-> (ARy /\ yRz)))
6		breq1 2672	. . . 4 |- (x = A -> (xRz <-> ARz))
7	5, 6	imbi12d 628	. . 3 |- (x = A -> (((xRy /\ yRz) -> xRz) <-> ((ARy /\ yRz) -> ARz)))
8		breq2 2673	. . . . 5 |- (y = B -> (ARy <-> ARB))
9		breq1 2672	. . . . 5 |- (y = B -> (yRz <-> BRz))
10	8, 9	anbi12d 630	. . . 4 |- (y = B -> ((ARy /\ yRz) <-> (ARB /\ BRz)))
11	10	imbi1d 615	. . 3 |- (y = B -> (((ARy /\ yRz) -> ARz) <-> ((ARB /\ BRz) -> ARz)))
12		breq2 2673	. . . . 5 |- (z = C -> (BRz <-> BRC))
13	12	anbi2d 618	. . . 4 |- (z = C -> ((ARB /\ BRz) <-> (ARB /\ BRC)))
14		breq2 2673	. . . 4 |- (z = C -> (ARz <-> ARC))
15	13, 14	imbi12d 628	. . 3 |- (z = C -> (((ARB /\ BRz) -> ARz) <-> ((ARB /\ BRC) -> ARC)))
16	7, 11, 15	syl3an9b 894	. 2 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
17		ertr.4	. . . . . . 7 |- Er R
18		dfer2 4346	. . . . . . 7 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
19	17, 18	mpbi 187	. . . . . 6 |- A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
20	19	a4i 1014	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
21	20	a4i 1014	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
22	21	a4i 1014	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
23	22	pm3.27i 322	. 2 |- ((xRy /\ yRz) -> xRz)
24	1, 2, 3, 16, 23	vtocl3 1882	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 3   /\ wa 221  A.wal 986   = wceq 988   e. wcel 990  Vcvv 1849   class class class wbr 2669  Er wer 4342

This theorem is referenced by:  erref 4359  erthi 4366  erdisj 4371  entr 4501

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-br 2670  df-opab 2718  df-cnv 3241  df-co 3242  df-er 4345

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