Theorem rexsns 3558

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 3539	. . . 4 |- ( x e. { A } <-> x = A )
2	1	anbi1i 453	. . 3 |- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii 1584	. 2 |- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex 2420	. 2 |- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5 2927	. 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 1331   E.wex 1468   e. wcel 1480   E.wrex 2415   [.wsbc 2904   {csn 3522

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-sbc 2905  df-sn 3528

This theorem is referenced by:  rexsng  3560  r19.12sn  3584  iunxsngf  3885  finexdc  6789  exfzdc  10010