Theorem ralsn 3675

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 3672	. 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 1372   e. wcel 2175   A.wral 2483   _Vcvv 2771   {csn 3632

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-sbc 2998  df-sn 3638

This theorem is referenced by:  tfr0dm  6407  elixpsn  6821  finomni  7241  eqs1  11080  nninfsellemdc  15909