Theorem erthi 6815

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 6778	. . 3 |- ( ph -> A e. X )
4	2, 3	erth 6813	. 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 1398   class class class wbr 4109   Er wer 6764   [cec 6765

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-er 6767  df-ec 6769

This theorem is referenced by:  qsel  6846  th3qlem1  6871  mulcanenqec  7701  mulcanenq0ec  7760  addnq0mo  7762  mulnq0mo  7763  addsrmo  8058  mulsrmo  8059  qusgrp2  13830  blpnfctr  15304