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Theorem elqsg 8665
Description: Closed form of elqs 8666. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
elqsg (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elqsg
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2741 . . 3 (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅𝐵 = [𝑥]𝑅))
21rexbidv 3173 . 2 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
3 df-qs 8612 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
42, 3elab2g 3630 1 (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wrex 3071  [cec 8604   / cqs 8605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rex 3072  df-qs 8612
This theorem is referenced by:  elqs  8666  elqsi  8667  elqsecl  8668  ecelqsg  8669  elpi1  24360  eldmqsres  36685  eldmqs1cossres  37059  prtlem11  37266
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