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Theorem elqsg 6058
Description: Closed form of elqs 6059. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Assertion
Ref Expression
elqsg (B 𝑉 → (B (A / 𝑅) ↔ x A 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 2022 . . 3 (y = B → (y = [x]𝑅B = [x]𝑅))
21rexbidv 2299 . 2 (y = B → (x A y = [x]𝑅x A B = [x]𝑅))
3 df-qs 6014 . 2 (A / 𝑅) = {yx A y = [x]𝑅}
42, 3elab2g 2660 1 (B 𝑉 → (B (A / 𝑅) ↔ x A B = [x]𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226   wcel 1369  wrex 2279  [cec 6006   / cqs 6007
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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1622  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-rex 2284  df-v 2531  df-qs 6014
This theorem is referenced by:  elqs  6059  elqsi  6060  ecelqsg  6061
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