Theorem erthi 7793

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 7753	. . 3 |- ( ph -> A e. X )
4	2, 3	erth 7791	. 2 |- ( ph -> ( A R B <-> [ A ] R = [ B ] R ) )
5	1, 4	mpbid 222	1 |- ( ph -> [ A ] R = [ B ] R )

Syntax hints:   -> wi 4   = wceq 1483   class class class wbr 4653   Er wer 7739   [cec 7740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-er 7742  df-ec 7744

This theorem is referenced by:  erdisj  7794  qsel  7826  addsrmo  9894  mulsrmo  9895  qusgrp2  17533  frgpinv  18177  qustgpopn  21923  blpnfctr  22241  pi1inv  22852  pi1xfrf  22853  pi1xfr  22855  pi1xfrcnvlem  22856  pi1cof  22859  vitalilem3  23379  sconnpi1  31221