Theorem ralsn 3650

Description: Convert a quantification over a singleton to a substitution. (Contributed by NM, 27-Apr-2009.)

Hypotheses

Ref	Expression
ralsn.1	|- A e. _V
ralsn.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ralsn	|- ( A. x e. { A } ph <-> ps )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem ralsn

Step	Hyp	Ref	Expression
1		ralsn.1	. 2 |- A e. _V
2		ralsn.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
3	2	ralsng 3647	. 2 |- ( A e. _V -> ( A. x e. { A } ph <-> ps ) )
4	1, 3	ax-mp 5	1 |- ( A. x e. { A } ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   {csn 3607

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 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-sbc 2978  df-sn 3613

This theorem is referenced by:  tfr0dm  6341  elixpsn  6753  finomni  7156  nninfsellemdc  15157