Theorem rexsn 3437

Description: Restricted existential quantification over a singleton. (Contributed by Jeff Madsen, 5-Jan-2011.)

Hypotheses

Ref	Expression
ralsn.1	|- A e. _V
ralsn.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

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

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem rexsn

Step	Hyp	Ref	Expression
1		ralsn.1	. 2 |- A e. _V
2		ralsn.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	rexsng 3434	. 2 |- ( A e. _V -> ( E. x e. { A } ph <-> ps ) )
4	1, 3	ax-mp 7	1 |- ( E. x e. { A } ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   _Vcvv 2601   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-sn 3404

This theorem is referenced by:  elsnres  4665  snec  6190  elreal  6997