Theorem rexsn 3575

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 3572	. 2 |- ( A e. _V -> ( E. x e. { A } ph <-> ps ) )
4	1, 3	ax-mp 5	1 |- ( E. x e. { A } ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   {csn 3532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-sn 3538

This theorem is referenced by:  elsnres  4864  snec  6498  0ct  7000  elreal  7660  restsn  12388