ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  erth2 Unicode version

Theorem erth2 6337
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 6306 . 2  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
3 erth2.2 . . . 4  |-  ( ph  ->  B  e.  X )
41, 3erth 6336 . . 3  |-  ( ph  ->  ( B R A  <->  [ B ] R  =  [ A ] R
) )
5 eqcom 2090 . . 3  |-  ( [ B ] R  =  [ A ] R  <->  [ A ] R  =  [ B ] R
)
64, 5syl6bb 194 . 2  |-  ( ph  ->  ( B R A  <->  [ A ] R  =  [ B ] R
) )
72, 6bitrd 186 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3845    Er wer 6289   [cec 6290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-er 6292  df-ec 6294
This theorem is referenced by:  qliftel  6372
  Copyright terms: Public domain W3C validator