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Theorem erth2 6062
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 (φ𝑅 Er 𝑋)
erth2.2 (φB 𝑋)
Assertion
Ref Expression
erth2 (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))

Proof of Theorem erth2
StepHypRef Expression
1 erth2.1 . . 3 (φ𝑅 Er 𝑋)
21ersymb 6031 . 2 (φ → (A𝑅BB𝑅A))
3 erth2.2 . . . 4 (φB 𝑋)
41, 3erth 6061 . . 3 (φ → (B𝑅A ↔ [B]𝑅 = [A]𝑅))
5 eqcom 2024 . . 3 ([B]𝑅 = [A]𝑅 ↔ [A]𝑅 = [B]𝑅)
64, 5syl6bb 185 . 2 (φ → (B𝑅A ↔ [A]𝑅 = [B]𝑅))
72, 6bitrd 177 1 (φ → (A𝑅B ↔ [A]𝑅 = [B]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374   class class class wbr 3738   Er wer 6014  [cec 6015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-er 6017  df-ec 6019
This theorem is referenced by:  qliftel  6097
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