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

Theorem spsbc 2893
Description: Specialization: if a formula is true for all sets, it is true for any class which is a set. Similar to Theorem 6.11 of [Quine] p. 44. See also stdpc4 1733 and rspsbc 2963. (Contributed by NM, 16-Jan-2004.)
Assertion
Ref Expression
spsbc  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)

Proof of Theorem spsbc
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 stdpc4 1733 . . . 4  |-  ( A. x ph  ->  [ y  /  x ] ph )
2 sbsbc 2886 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylib 121 . . 3  |-  ( A. x ph  ->  [. y  /  x ]. ph )
4 dfsbcq 2884 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
53, 4syl5ib 153 . 2  |-  ( y  =  A  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
65vtocleg 2731 1  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314    = wceq 1316    e. wcel 1465   [wsb 1720   [.wsbc 2882
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-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662  df-sbc 2883
This theorem is referenced by:  spsbcd  2894  sbcth  2895  sbcthdv  2896  sbceqal  2936  sbcimdv  2946  csbiebt  3009  csbexga  4026
  Copyright terms: Public domain W3C validator