Theorem ertr 3211

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 2065	. . . . 5 |- (x = A -> (xRy <-> ARy))
5	4	anbi1d 469	. . . 4 |- (x = A -> ((xRy /\ yRz) <-> (ARy /\ yRz)))
6		breq1 2065	. . . 4 |- (x = A -> (xRz <-> ARz))
7	5, 6	imbi12d 474	. . 3 |- (x = A -> (((xRy /\ yRz) -> xRz) <-> ((ARy /\ yRz) -> ARz)))
8		breq2 2066	. . . . 5 |- (y = B -> (ARy <-> ARB))
9		breq1 2065	. . . . 5 |- (y = B -> (yRz <-> BRz))
10	8, 9	anbi12d 476	. . . 4 |- (y = B -> ((ARy /\ yRz) <-> (ARB /\ BRz)))
11	10	imbi1d 465	. . 3 |- (y = B -> (((ARy /\ yRz) -> ARz) <-> ((ARB /\ BRz) -> ARz)))
12		breq2 2066	. . . . 5 |- (z = C -> (BRz <-> BRC))
13	12	anbi2d 468	. . . 4 |- (z = C -> ((ARB /\ BRz) <-> (ARB /\ BRC)))
14		breq2 2066	. . . 4 |- (z = C -> (ARz <-> ARC))
15	13, 14	imbi12d 474	. . 3 |- (z = C -> (((ARB /\ BRz) -> ARz) <-> ((ARB /\ BRC) -> ARC)))
16	7, 11, 15	syl3an9b 634	. 2 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
17		ertr.4	. . . . . . 7 |- Er R
18		er2 3201	. . . . . . 7 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
19	17, 18	mpbi 164	. . . . . 6 |- A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
20	19	a4i 680	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
21	20	a4i 680	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
22	21	a4i 680	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
23	22	pm3.27i 261	. 2 |- ((xRy /\ yRz) -> xRz)
24	1, 2, 3, 16, 23	vtocl3 1380	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  Vcvv 1348   class class class wbr 2054  Er wer 3197

This theorem is referenced by:  erref 3212  erthi 3218  erdisj 3223  entrt 3319

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-co 2427  df-er 3200

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