Theorem rexsng 4007

Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)

Hypothesis

Ref	Expression
ralsng.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rexsng	|- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem rexsng

Step	Hyp	Ref	Expression
1		rexsns 4004	. 2 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3300	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	syl5bb 261	1 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 188   = wceq 1444   e. wcel 1887   E.wrex 2738   [.wsbc 3267   {csn 3968

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-rex 2743  df-v 3047  df-sbc 3268  df-sn 3969

This theorem is referenced by:  rexsn  4011  rexprg  4022  rextpg  4024  iunxsng  4360  frirr  4811  frsn  4905  imasng  5190  scshwfzeqfzo  12925  mnd1  16577  mnd1OLD  16578  grp1  16758  frgra2v  25727  1vwmgra  25731  ballotlemfc0  29325  ballotlemfcc  29326  elpaddat  33369  elpadd2at  33371  brfvidRP  36280  iccelpart  38747  zlidlring  39981  lco0  40273