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.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. . 3 ⊢ (y = B → (y = [x]R ↔ B = [x]R))
2	1	rexbidv 2635	. 2 ⊢ (y = B → (∃x ∈ A y = [x]R ↔ ∃x ∈ A B = [x]R))
3		df-qs 5951	. 2 ⊢ (A / R) = {y ∣ ∃x ∈ A y = [x]R}
4	2, 3	elab2g 2987	1 ⊢ (B ∈ V → (B ∈ (A / R) ↔ ∃x ∈ A B = [x]R))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  [cec 5945   / cqs 5946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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