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Theorem erthi 6942
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
StepHypRef Expression
1 erthi.2 . 2  |-  ( ph  ->  A R B )
2 erthi.1 . . 3  |-  ( ph  ->  R  Er  X )
32, 1ercl 6907 . . 3  |-  ( ph  ->  A  e.  X )
42, 3erth 6940 . 2  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
51, 4mpbid 202 1  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   class class class wbr 4204    Er wer 6893   [cec 6894
This theorem is referenced by:  erdisj  6943  qsel  6974  th3qlem1  7001  divsgrp2  14924  frgpinv  15384  divstgpopn  18137  blpnfctr  18454  pi1inv  19065  pi1xfrf  19066  pi1xfr  19068  pi1xfrcnvlem  19069  pi1cof  19072  vitalilem3  19490  sconpi1  24914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-er 6896  df-ec 6898
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