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Theorem erth2 6941
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 6910 . 2  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
3 erth2.2 . . . 4  |-  ( ph  ->  B  e.  X )
41, 3erth 6940 . . 3  |-  ( ph  ->  ( B R A  <->  [ B ] R  =  [ A ] R
) )
5 eqcom 2437 . . 3  |-  ( [ B ] R  =  [ A ] R  <->  [ A ] R  =  [ B ] R
)
64, 5syl6bb 253 . 2  |-  ( ph  ->  ( B R A  <->  [ A ] R  =  [ B ] R
) )
72, 6bitrd 245 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   class class class wbr 4204    Er wer 6893   [cec 6894
This theorem is referenced by:  qliftel  6978
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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