Theorem rexsns 3744

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 3722	. . . 4 |- ( x e. { A } <-> x = A )
2	1	anbi1i 462	. . 3 |- ( ( x e. { A } /\ ph ) <-> ( x = A /\ ph ) )
3	2	exbii 1658	. 2 |- ( E. x ( x e. { A } /\ ph ) <-> E. x ( x = A /\ ph ) )
4		df-rex 2534	. 2 |- ( E. x e. { A } ph <-> E. x ( x e. { A } /\ ph ) )
5		sbc5 3075	. 2 |- ( [. A / x ]. ph <-> E. x ( x = A /\ ph ) )
6	3, 4, 5	3bitr4i 212	1 |- ( E. x e. { A } ph <-> [. A / x ]. ph )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   [.wsbc 3051   {csn 3705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-sn 3711

This theorem is referenced by:  rexsng  3746  r19.12sn  3771  iunxsngf  4085  finexdc  7197  exfzdc  10637