Theorem elqsg 7798

Description: Closed form of elqs 7799. (Contributed by Rodolfo Medina, 12-Oct-2010.)

Assertion

Ref	Expression
elqsg	|- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. 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 2626	. . 3 |- ( y = B -> ( y = [ x ] R <-> B = [ x ] R ) )
2	1	rexbidv 3052	. 2 |- ( y = B -> ( E. x e. A y = [ x ] R <-> E. x e. A B = [ x ] R ) )
3		df-qs 7748	. 2 |- ( A /. R ) = { y | E. x e. A y = [ x ] R }
4	2, 3	elab2g 3353	1 |- ( B e. V -> ( B e. ( A /. R ) <-> E. x e. A B = [ x ] R ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E.wrex 2913   [cec 7740   /.cqs 7741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-qs 7748

This theorem is referenced by:  elqs  7799  elqsi  7800  elqsecl  7801  ecelqsg  7802  elpi1  22845  eldmqsres  34051  prtlem11  34151