Theorem ralsns 3597

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 2440	. . 3 |- ( A. x e. { A } ph <-> A. x ( x e. { A } -> ph ) )
2		velsn 3577	. . . . 5 |- ( x e. { A } <-> x = A )
3	2	imbi1i 237	. . . 4 |- ( ( x e. { A } -> ph ) <-> ( x = A -> ph ) )
4	3	albii 1450	. . 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 2961	. 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 1333   = wceq 1335   e. wcel 2128   A.wral 2435   [.wsbc 2937   {csn 3560

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-sbc 2938  df-sn 3566

This theorem is referenced by:  ralsng  3599  sbcsng  3618  rabrsndc  3627  omsinds  4580  ssfirab  6875  uzsinds  10334