Theorem ralsnsg 3430

Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
ralsnsg	|- ( 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 ralsnsg

Step	Hyp	Ref	Expression
1		sbc6g 2839	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
2		df-ral 2353	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
3		velsn 3415	. . . . 5 |- ( x e. { A } <-> x = A )
4	3	imbi1i 236	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
5	4	albii 1399	. . 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 1282   = wceq 1284   e. wcel 1433   A.wral 2348   [.wsbc 2815   {csn 3398

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816  df-sn 3404

This theorem is referenced by:  ac6sfi  6379  rexfiuz  9875  prmind2  10502