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Theorem erth2 6721
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 6690 . 2  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
3 erth2.2 . . . 4  |-  ( ph  ->  B  e.  X )
41, 3erth 6720 . . 3  |-  ( ph  ->  ( B R A  <->  [ B ] R  =  [ A ] R
) )
5 eqcom 2298 . . 3  |-  ( [ B ] R  =  [ A ] R  <->  [ A ] R  =  [ B ] R
)
64, 5syl6bb 252 . 2  |-  ( ph  ->  ( B R A  <->  [ A ] R  =  [ B ] R
) )
72, 6bitrd 244 1  |-  ( ph  ->  ( A R B  <->  [ A ] R  =  [ B ] R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   class class class wbr 4039    Er wer 6673   [cec 6674
This theorem is referenced by:  qliftel  6757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678
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