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Theorem ercl2 8713
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ercl2 (𝜑𝐵𝑋)

Proof of Theorem ercl2
StepHypRef Expression
1 ersym.1 . 2 (𝜑𝑅 Er 𝑋)
2 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
31, 2ersym 8712 . 2 (𝜑𝐵𝑅𝐴)
41, 3ercl 8711 1 (𝜑𝐵𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5139   Er wer 8697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-er 8700
This theorem is referenced by:  qliftfun  8793  efgcpbl2  19673  frgpcpbl  19675  prjspner1  41918
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