Theorem rexsns 3615

Description: Restricted existential quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.) (Revised by NM, 22-Aug-2018.)

Assertion

Ref	Expression
rexsns	|- ( E. x e. { A } ph <-> [. A / x ]. ph )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem rexsns

Step	Hyp	Ref	Expression
1		velsn 3593	. . . 4 |- ( x e. { A } <-> x = A )
2	1	anbi1i 454	. . 3 |- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii 1593	. 2 |- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex 2450	. 2 |- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5 2974	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
6	3, 4, 5	3bitr4i 211	1 |- ( E. x e. { A } ph <-> [. A / x ]. ph )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1343   E.wex 1480   e. wcel 2136   E.wrex 2445   [.wsbc 2951   {csn 3576

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-sn 3582

This theorem is referenced by:  rexsng  3617  r19.12sn  3642  iunxsngf  3943  finexdc  6868  exfzdc  10175