Theorem ralsn 3619

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 3616	. 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 104   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   {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-3an 970  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:  tfr0dm  6290  elixpsn  6701  finomni  7104  nninfsellemdc  13890