Theorem ralsns 3614

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		df-ral 2449	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
2		velsn 3593	. . . . 5 |- ( x e. { A } <-> x = A )
3	2	imbi1i 237	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
4	3	albii 1458	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
5	1, 4	bitri 183	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
6		sbc6g 2975	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
7	5, 6	bitr4id 198	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   [.wsbc 2951   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-sbc 2952  df-sn 3582

This theorem is referenced by:  ralsng  3616  sbcsng  3635  rabrsndc  3644  omsinds  4599  ssfirab  6899  uzsinds  10377