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Theorem erth2 6701
Description: Basic property of equivalence relations. Compare Theorem 73 of [Suppes] p. 82. Assumes membership of the second argument in the domain. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)
Hypotheses
Ref Expression
erth2.1  |-  ( ph  ->  R  Er  X )
erth2.2  |-  ( ph  ->  B  e.  X )
Assertion
Ref Expression
erth2  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )

Proof of Theorem erth2
StepHypRef Expression
1 erth2.1 . . 3  |-  ( ph  ->  R  Er  X )
21ersymb 6670 . 2  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
3 erth2.2 . . . 4  |-  ( ph  ->  B  e.  X )
41, 3erth 6700 . . 3  |-  ( ph  ->  ( B R A  <->  [ B ] R  =  [ A ] R
) )
5 eqcom 2287 . . 3  |-  ( [ B ] R  =  [ A ] R  <->  [ A ] R  =  [ B ] R
)
64, 5syl6bb 254 . 2  |-  ( ph  ->  ( B R A  <->  [ A ] R  =  [ B ] R
) )
72, 6bitrd 246 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1624    e. wcel 1685   class class class wbr 4025    Er wer 6653   [cec 6654
This theorem is referenced by:  qliftel  6737
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6656  df-ec 6658
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