Theorem rexsn 4223

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 4219	. 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 196   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-sn 4178

This theorem is referenced by:  elsnres  5436  oarec  7642  snec  7810  zornn0g  9327  fpwwe2lem13  9464  elreal  9952  hashge2el2difr  13263  vdwlem6  15690  pmatcollpw3fi1  20593  restsn  20974  snclseqg  21919  ust0  22023  esum2dlem  30154  eulerpartlemgh  30440  eldm3  31651  poimirlem28  33437  heiborlem3  33612