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Theorem elqsg 5976
 Description: Closed form of elqs 5977. (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 2359 . . 3 (y = B → (y = [x]RB = [x]R))
21rexbidv 2635 . 2 (y = B → (x A y = [x]Rx A B = [x]R))
3 df-qs 5951 . 2 (A / R) = {y x A y = [x]R}
42, 3elab2g 2987 1 (B V → (B (A / R) ↔ x A B = [x]R))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  [cec 5945   / cqs 5946 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  elqs  5977  elqsi  5978  ecelqsg  5979  elncs  6119
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