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Theorem erthi 6910
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 6875 . . 3  |-  ( ph  ->  A  e.  X )
42, 3erth 6908 . 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 1649   class class class wbr 4172    Er wer 6861   [cec 6862
This theorem is referenced by:  erdisj  6911  qsel  6942  th3qlem1  6969  divsgrp2  14891  frgpinv  15351  divstgpopn  18102  blpnfctr  18419  pi1inv  19030  pi1xfrf  19031  pi1xfr  19033  pi1xfrcnvlem  19034  pi1cof  19037  vitalilem3  19455  sconpi1  24879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-er 6864  df-ec 6866
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