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Theorem erth2 6591
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 6560 . 2  |-  ( ph  ->  ( A R B  <-> 
B R A ) )
3 erth2.2 . . . 4  |-  ( ph  ->  B  e.  X )
41, 3erth 6590 . . 3  |-  ( ph  ->  ( B R A  <->  [ B ] R  =  [ A ] R
) )
5 eqcom 2255 . . 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 1619    e. wcel 1621   class class class wbr 3920    Er wer 6543   [cec 6544
This theorem is referenced by:  qliftel  6627
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-er 6546  df-ec 6548
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