Theorem erthi 4281

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 4273	. . . . 5 |- (ARB <-> BRA)
5		visset 1813	. . . . . . 7 |- x e. V
6	2, 1, 5, 3	ertr 4274	. . . . . 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 4274	. . . . 5 |- ((ARB /\ BRx) -> ARx)
10	9	ex 373	. . . 4 |- (ARB -> (BRx -> ARx))
11	8, 10	impbid 516	. . 3 |- (ARB -> (ARx <-> BRx))
12	5, 1	elec 4279	. . 3 |- (x e. [A]R <-> ARx)
13	5, 2	elec 4279	. . 3 |- (x e. [B]R <-> BRx)
14	11, 12, 13	3bitr4g 555	. 2 |- (ARB -> (x e. [A]R <-> x e. [B]R))
15	14	eqrdv 1473	1 |- (ARB -> [A]R = [B]R)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  Vcvv 1811   class class class wbr 2619  Er wer 4258  [cec 4259

This theorem is referenced by:  erth 4282  erdisj 4286  th3qlem1 4314  distrpqlem 5066

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-er 4261  df-ec 4263

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