Theorem rexsng 3684

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 3682	. 2 |- ( E. x e. { A } ph <-> [. A / x ]. ph )
2		ralsng.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	sbcieg 3038	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> ps ) )
4	1, 3	bitrid 192	1 |- ( A e. V -> ( E. x e. { A } ph <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   E.wrex 2487   [.wsbc 3005   {csn 3643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-sbc 3006  df-sn 3649

This theorem is referenced by:  rexsn  3687  rexprg  3695  rextpg  3697  iunxsng  4017  imasng  5066  dvdsprmpweqnn  12774  mnd1  13402  grp1  13553