Theorem erthi 4287

Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57.

Hypotheses

Ref	Expression
erthi.1	|- A e. V
erthi.2	|- B e. V
erthi.3	|- Er R

Assertion

Ref	Expression
erthi	|- (ARB -> [A]R = [B]R)

Proof of Theorem erthi

Step	Hyp	Ref	Expression
1		erthi.1	. . . . . 6 |- A e. V
2		erthi.2	. . . . . 6 |- B e. V
3		erthi.3	. . . . . 6 |- Er R
4	1, 2, 3	ersymb 4279	. . . . 5 |- (ARB <-> BRA)
5		visset 1816	. . . . . . 7 |- x e. V
6	2, 1, 5, 3	ertr 4280	. . . . . 6 |- ((BRA /\ ARx) -> BRx)
7	6	ex 373	. . . . 5 |- (BRA -> (ARx -> BRx))
8	4, 7	sylbi 199	. . . 4 |- (ARB -> (ARx -> BRx))
9	1, 2, 5, 3	ertr 4280	. . . . 5 |- ((ARB /\ BRx) -> ARx)
10	9	ex 373	. . . 4 |- (ARB -> (BRx -> ARx))
11	8, 10	impbid 518	. . 3 |- (ARB -> (ARx <-> BRx))
12	5, 1	elec 4285	. . 3 |- (x e. [A]R <-> ARx)
13	5, 2	elec 4285	. . 3 |- (x e. [B]R <-> BRx)
14	11, 12, 13	3bitr4g 557	. 2 |- (ARB -> (x e. [A]R <-> x e. [B]R))
15	14	eqrdv 1476	1 |- (ARB -> [A]R = [B]R)

Syntax hints:   -> wi 3   = wceq 958   e. wcel 960  Vcvv 1814   class class class wbr 2624  Er wer 4264  [cec 4265

This theorem is referenced by:  erth 4288  erdisj 4292  th3qlem1 4320  distrpqlem 5078

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-er 4267  df-ec 4269

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