Theorem ertr 4271

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 2619	. . . . 5 |- (x = A -> (xRy <-> ARy))
5	4	anbi1d 616	. . . 4 |- (x = A -> ((xRy /\ yRz) <-> (ARy /\ yRz)))
6		breq1 2619	. . . 4 |- (x = A -> (xRz <-> ARz))
7	5, 6	imbi12d 625	. . 3 |- (x = A -> (((xRy /\ yRz) -> xRz) <-> ((ARy /\ yRz) -> ARz)))
8		breq2 2620	. . . . 5 |- (y = B -> (ARy <-> ARB))
9		breq1 2619	. . . . 5 |- (y = B -> (yRz <-> BRz))
10	8, 9	anbi12d 627	. . . 4 |- (y = B -> ((ARy /\ yRz) <-> (ARB /\ BRz)))
11	10	imbi1d 612	. . 3 |- (y = B -> (((ARy /\ yRz) -> ARz) <-> ((ARB /\ BRz) -> ARz)))
12		breq2 2620	. . . . 5 |- (z = C -> (BRz <-> BRC))
13	12	anbi2d 615	. . . 4 |- (z = C -> ((ARB /\ BRz) <-> (ARB /\ BRC)))
14		breq2 2620	. . . 4 |- (z = C -> (ARz <-> ARC))
15	13, 14	imbi12d 625	. . 3 |- (z = C -> (((ARB /\ BRz) -> ARz) <-> ((ARB /\ BRC) -> ARC)))
16	7, 11, 15	syl3an9b 890	. 2 |- ((x = A /\ y = B /\ z = C) -> (((xRy /\ yRz) -> xRz) <-> ((ARB /\ BRC) -> ARC)))
17		ertr.4	. . . . . . 7 |- Er R
18		dfer2 4259	. . . . . . 7 |- (Er R <-> A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz)))
19	17, 18	mpbi 189	. . . . . 6 |- A.xA.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
20	19	a4i 981	. . . . 5 |- A.yA.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
21	20	a4i 981	. . . 4 |- A.z((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
22	21	a4i 981	. . 3 |- ((xRy -> yRx) /\ ((xRy /\ yRz) -> xRz))
23	22	pm3.27i 324	. 2 |- ((xRy /\ yRz) -> xRz)
24	1, 2, 3, 16, 23	vtocl3 1842	1 |- ((ARB /\ BRC) -> ARC)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   = wceq 955   e. wcel 957  Vcvv 1809   class class class wbr 2616  Er wer 4255

This theorem is referenced by:  erref 4272  erthi 4278  erdisj 4283  entrt 4408

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 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-cnv 3183  df-co 3184  df-er 4258

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