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Theorem elqsg 6479
Description: Closed form of elqs 6480. (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 2146 . . 3 (𝑦 = 𝐵 → (𝑦 = [𝑥]𝑅𝐵 = [𝑥]𝑅))
21rexbidv 2438 . 2 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝑅 ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
3 df-qs 6435 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
42, 3elab2g 2831 1 (𝐵𝑉 → (𝐵 ∈ (𝐴 / 𝑅) ↔ ∃𝑥𝐴 𝐵 = [𝑥]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1331  wcel 1480  wrex 2417  [cec 6427   / cqs 6428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-qs 6435
This theorem is referenced by:  elqs  6480  elqsi  6481  ecelqsg  6482
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