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Theorem elqs 5977
 Description: Membership in a quotient set. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 12-Nov-2008.)
Hypothesis
Ref Expression
elqs.1 B V
Assertion
Ref Expression
elqs (B (A / R) ↔ x A B = [x]R)
Distinct variable groups:   x,A   x,B   x,R

Proof of Theorem elqs
StepHypRef Expression
1 elqs.1 . 2 B V
2 elqsg 5976 . 2 (B V → (B (A / R) ↔ x A B = [x]R))
31, 2ax-mp 5 1 (B (A / R) ↔ x A B = [x]R)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-v 2861  df-qs 5951 This theorem is referenced by:  qsid  5990
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