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Theorem ersymtr 5932
 Description: Equivalence relationship as symmetric, transitive relationship. (Contributed by SF, 22-Feb-2015.)
Assertion
Ref Expression
ersymtr (R Er A ↔ (R Sym A R Trans A))

Proof of Theorem ersymtr
StepHypRef Expression
1 df-er 5909 . . 3 Er = ( SymTrans )
21breqi 4645 . 2 (R Er AR( SymTrans )A)
3 brin 4693 . 2 (R( SymTrans )A ↔ (R Sym A R Trans A))
42, 3bitri 240 1 (R Er A ↔ (R Sym A R Trans A))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∩ cin 3208   class class class wbr 4639   Trans ctrans 5888   Sym csym 5897   Er cer 5898 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-br 4640  df-er 5909 This theorem is referenced by:  iserd  5942  ersym  5952  ertr  5954  ertrd  5955
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