Theorem erthi 6845

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 6808	. . 3 |- ( ph -> A e. X )
4	2, 3	erth 6843	. 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 1402   class class class wbr 4125   Er wer 6794   [cec 6795

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-er 6797  df-ec 6799

This theorem is referenced by:  qsel  6876  th3qlem1  6901  mulcanenqec  7743  mulcanenq0ec  7802  addnq0mo  7804  mulnq0mo  7805  addsrmo  8100  mulsrmo  8101  qusgrp2  13893  blpnfctr  15463