Theorem rexsn 3627

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 3624	. 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 1348   e. wcel 2141   E.wrex 2449   _Vcvv 2730   {csn 3583

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-sn 3589

This theorem is referenced by:  elsnres  4928  snec  6574  0ct  7084  elreal  7790  restsn  12974