Theorem ralsns 3656

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 2477	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
2		velsn 3635	. . . . 5 |- ( x e. { A } <-> x = A )
3	2	imbi1i 238	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
4	3	albii 1481	. . 3 |- ( A. x ( x e. { A } -> ph ) <-> A. x ( x = A -> ph ) )
5	1, 4	bitri 184	. 2 |- ( A. x e. { A } ph <-> A. x ( x = A -> ph ) )
6		sbc6g 3010	. 2 |- ( A e. V -> ( [. A / x ]. ph <-> A. x ( x = A -> ph ) ) )
7	5, 6	bitr4id 199	1 |- ( A e. V -> ( A. x e. { A } ph <-> [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 105   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   [.wsbc 2985   {csn 3618

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-sbc 2986  df-sn 3624

This theorem is referenced by:  ralsng  3658  sbcsng  3677  rabrsndc  3686  omsinds  4654  ssfirab  6990  dcfi  7040  uzsinds  10515