Theorem rexsns 3409

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 3392	. . . 4 |- ( x e. { A } <-> x = A )
2	1	anbi1i 431	. . 3 |- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii 1496	. 2 |- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex 2312	. 2 |- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5 2787	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
6	3, 4, 5	3bitr4i 201	1 |- ( E. x e. { A } ph <-> [. A / x ]. ph )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   [.wsbc 2764   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-sn 3381

This theorem is referenced by: (None)