MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralsn Unicode version

Theorem ralsn 3809
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
StepHypRef Expression
1 ralsn.1 . 2  |-  A  e. 
_V
2 ralsn.2 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32ralsng 3806 . 2  |-  ( A  e.  _V  ->  ( A. x  e.  { A } ph  <->  ps ) )
41, 3ax-mp 8 1  |-  ( A. x  e.  { A } ph  <->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   {csn 3774
This theorem is referenced by:  elixpsn  7060  frfi  7311  dffi3  7394  fseqenlem1  7861  fpwwe2lem13  8473  hashbc  11657  hashf1lem1  11659  rpnnen2lem11  12779  drsdirfi  14350  0subg  14920  efgsp1  15324  dprd2da  15555  lbsextlem4  16188  txkgen  17637  xkoinjcn  17672  isufil2  17893  ust0  18202  prdsxmetlem  18351  prdsbl  18474  finiunmbl  19391  xrlimcnp  20760  chtub  20949  2sqlem10  21111  dchrisum0flb  21157  pntpbnd1  21233  usgra1v  21362  constr1trl  21541  h1deoi  23004  subfacp1lem5  24823  cvmlift2lem1  24942  cvmlift2lem12  24954  heibor1lem  26408  bnj149  28952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-v 2918  df-sbc 3122  df-sn 3780
  Copyright terms: Public domain W3C validator