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Theorem erthi 6701
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 6666 . . 3  |-  ( ph  ->  A  e.  X )
42, 3erth 6699 . 2  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
51, 4mpbid 203 1  |-  ( ph  ->  [ A ] R  =  [ B ] R
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1628   class class class wbr 4024    Er wer 6652   [cec 6653
This theorem is referenced by:  erdisj  6702  qsel  6733  th3qlem1  6759  divsgrp2  14607  frgpinv  15067  divstgpopn  17796  blpnfctr  17976  pi1inv  18544  pi1xfrf  18545  pi1xfr  18547  pi1xfrcnvlem  18548  pi1cof  18551  vitalilem3  18959  sconpi1  23174  pdiveql  25567
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-er 6655  df-ec 6657
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