Theorem ralsns 3439

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by Mario Carneiro, 23-Apr-2015.)

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   V( x)

Proof of Theorem ralsns

Step	Hyp	Ref	Expression
1		sbc6g 2840	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral 2354	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		velsn 3423	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	imbi1i 236	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii 1400	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
6	2, 5	bitri 182	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
7	1, 6	syl6rbbr 197	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 103   A.wal 1283   = wceq 1285   e. wcel 1434   A.wral 2349   [.wsbc 2816   {csn 3406

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-sbc 2817  df-sn 3412

This theorem is referenced by:  ralsng  3441  sbcsng  3459  rabrsndc  3468  uzsinds  9518