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Theorem erth2 6087
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 6056 . 2 (φ → (A𝑅BB𝑅A))
3 erth2.2 . . . 4 (φB 𝑋)
41, 3erth 6086 . . 3 (φ → (B𝑅A ↔ [B]𝑅 = [A]𝑅))
5 eqcom 2039 . . 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 1242   wcel 1390   class class class wbr 3755   Er wer 6039  [cec 6040
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044
This theorem is referenced by:  qliftel  6122
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