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Theorem elqsg 8365
 Description: Closed form of elqs 8366. (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 2763 . . 3 (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅𝐵 = [𝑥]𝑅))
21rexbidv 3222 . 2 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
3 df-qs 8312 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
42, 3elab2g 3592 1 (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  [cec 8304   / cqs 8305 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-rex 3077  df-qs 8312 This theorem is referenced by:  elqs  8366  elqsi  8367  elqsecl  8368  ecelqsg  8369  elpi1  23761  eldmqsres  36019  eldmqs1cossres  36369  prtlem11  36478
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