Theorem rexsng 3998

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 3996	. 2 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3288	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	syl5bb 265	1 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   E.wrex 2757   [.wsbc 3255   {csn 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-rex 2762  df-v 3033  df-sbc 3256  df-sn 3960

This theorem is referenced by:  rexsn  4002  rexprg  4013  rextpg  4015  iunxsng  4351  frirr  4816  frsn  4910  imasng  5196  scshwfzeqfzo  12982  mnd1  16655  mnd1OLD  16656  grp1  16836  frgra2v  25806  1vwmgra  25810  ballotlemfc0  29398  ballotlemfcc  29399  elpaddat  33440  elpadd2at  33442  brfvidRP  36351  iccelpart  38892  1loopgrvd0  39726  1egrvtxdg0  39734  zlidlring  40436  lco0  40728