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Theorem erthi 6674
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 6639 . . 3  |-  ( ph  ->  A  e.  X )
42, 3erth 6672 . 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 1619   class class class wbr 3997    Er wer 6625   [cec 6626
This theorem is referenced by:  erdisj  6675  qsel  6706  th3qlem1  6732  divsgrp2  14575  frgpinv  15035  divstgpopn  17764  blpnfctr  17944  pi1inv  18512  pi1xfrf  18513  pi1xfr  18515  pi1xfrcnvlem  18516  pi1cof  18519  vitalilem3  18927  sconpi1  23142  pdiveql  25535
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-er 6628  df-ec 6630
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