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Theorem erth2 8796
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 (𝜑𝐵𝑋)
Assertion
Ref Expression
erth2 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem erth2
StepHypRef Expression
1 erth2.1 . . 3 (𝜑𝑅 Er 𝑋)
21ersymb 8758 . 2 (𝜑 → (𝐴𝑅𝐵𝐵𝑅𝐴))
3 erth2.2 . . . 4 (𝜑𝐵𝑋)
41, 3erth 8795 . . 3 (𝜑 → (𝐵𝑅𝐴 ↔ [𝐵]𝑅 = [𝐴]𝑅))
5 eqcom 2742 . . 3 ([𝐵]𝑅 = [𝐴]𝑅 ↔ [𝐴]𝑅 = [𝐵]𝑅)
64, 5bitrdi 287 . 2 (𝜑 → (𝐵𝑅𝐴 ↔ [𝐴]𝑅 = [𝐵]𝑅))
72, 6bitrd 279 1 (𝜑 → (𝐴𝑅𝐵 ↔ [𝐴]𝑅 = [𝐵]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106   class class class wbr 5148   Er wer 8741  [cec 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-er 8744  df-ec 8746
This theorem is referenced by:  qliftel  8839  qusker  33357  qsdrnglem2  33504
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