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Theorem elqsi 6094
 Description: Membership in a quotient set. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
elqsi (B (A / 𝑅) → x A B = [x]𝑅)
Distinct variable groups:   x,A   x,B   x,𝑅

Proof of Theorem elqsi
StepHypRef Expression
1 elqsg 6092 . 2 (B (A / 𝑅) → (B (A / 𝑅) ↔ x A B = [x]𝑅))
21ibi 165 1 (B (A / 𝑅) → x A B = [x]𝑅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  [cec 6040   / cqs 6041 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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-qs 6048 This theorem is referenced by:  ectocld  6108  ecoptocl  6129  eroveu  6133  dmaddpqlem  6361  nqpi  6362  nq0nn  6424
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