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

Theorem ralsnsg 4014
Description: Substitution expressed in terms of quantification over a singleton. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
ralsnsg  |-  ( 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 ralsnsg
StepHypRef Expression
1 sbc6g 3304 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  A. x ( x  =  A  ->  ph )
) )
2 df-ral 2753 . . 3  |-  ( A. x  e.  { A } ph  <->  A. x ( x  e.  { A }  ->  ph ) )
3 elsn 3993 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43imbi1i 331 . . . 4  |-  ( ( x  e.  { A }  ->  ph )  <->  ( x  =  A  ->  ph )
)
54albii 1701 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  ph )  <->  A. x ( x  =  A  ->  ph )
)
62, 5bitri 257 . 2  |-  ( A. x  e.  { A } ph  <->  A. x ( x  =  A  ->  ph )
)
71, 6syl6rbbr 272 1  |-  ( A  e.  V  ->  ( A. x  e.  { A } ph  <->  [. A  /  x ]. ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189   A.wal 1452    = wceq 1454    e. wcel 1897   A.wral 2748   [.wsbc 3278   {csn 3979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-ral 2753  df-v 3058  df-sbc 3279  df-sn 3980
This theorem is referenced by:  ralsng  4017  ixpsnval  7550  ac6sfi  7840  rexfiuz  13458  prmind2  14683  finixpnum  31974
  Copyright terms: Public domain W3C validator