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Theorem ercl2 7642
 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 7641 . 2 (𝜑𝐵𝑅𝐴)
41, 3ercl 7640 1 (𝜑𝐵𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   class class class wbr 4583   Er wer 7626 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-er 7629 This theorem is referenced by:  qliftfun  7719  efgcpbl2  17993  frgpcpbl  17995
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