Theorem erthi 6637

Description: Basic property of equivalence relations. Part of Lemma 3N of [Enderton] p. 57. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
erthi.1	|- ( ph -> R Er X )
erthi.2	|- ( ph -> A R B )

Assertion

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

Proof of Theorem erthi

Step	Hyp	Ref	Expression
1		erthi.2	. 2 |- ( ph -> A R B )
2		erthi.1	. . 3 |- ( ph -> R Er X )
3	2, 1	ercl 6600	. . 3 |- ( ph -> A e. X )
4	2, 3	erth 6635	. 2 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )
5	1, 4	mpbid 147	1 |- ( ph -> [ A ] R = [ B ] R )

Syntax hints:   -> wi 4   = wceq 1364   class class class wbr 4030   Er wer 6586   [cec 6587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-er 6589  df-ec 6591

This theorem is referenced by:  qsel  6668  th3qlem1  6693  mulcanenqec  7448  mulcanenq0ec  7507  addnq0mo  7509  mulnq0mo  7510  addsrmo  7805  mulsrmo  7806  qusgrp2  13186  blpnfctr  14618