ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rexsng Unicode version

Theorem rexsng 3624
Description: Restricted existential quantification over a singleton. (Contributed by NM, 29-Jan-2012.)
Hypothesis
Ref Expression
ralsng.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
rexsng  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
Distinct variable groups:    x, A    ps, x
Allowed substitution hints:    ph( x)    V( x)

Proof of Theorem rexsng
StepHypRef Expression
1 rexsns 3622 . 2  |-  ( E. x  e.  { A } ph  <->  [. A  /  x ]. ph )
2 ralsng.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
32sbcieg 2987 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ps ) )
41, 3syl5bb 191 1  |-  ( A  e.  V  ->  ( E. x  e.  { A } ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1348    e. wcel 2141   E.wrex 2449   [.wsbc 2955   {csn 3583
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sbc 2956  df-sn 3589
This theorem is referenced by:  rexsn  3627  rexprg  3635  rextpg  3637  iunxsng  3948  imasng  4976  dvdsprmpweqnn  12289  mnd1  12679
  Copyright terms: Public domain W3C validator