Theorem erthi 6749

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 6712	. . 3 |- ( ph -> A e. X )
4	2, 3	erth 6747	. 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 1397   class class class wbr 4088   Er wer 6698   [cec 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-er 6701  df-ec 6703

This theorem is referenced by:  qsel  6780  th3qlem1  6805  mulcanenqec  7605  mulcanenq0ec  7664  addnq0mo  7666  mulnq0mo  7667  addsrmo  7962  mulsrmo  7963  qusgrp2  13699  blpnfctr  15162