Theorem rexsns 3486

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 3467	. . . 4 |- ( x e. { A } <-> x = A )
2	1	anbi1i 447	. . 3 |- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii 1542	. 2 |- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex 2366	. 2 |- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5 2864	. 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 1290   E.wex 1427   e. wcel 1439   E.wrex 2361   [.wsbc 2841   {csn 3450

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-sbc 2842  df-sn 3456

This theorem is referenced by:  rexsng  3488  r19.12sn  3512  iunxsngf  3813  finexdc  6672  exfzdc  9705