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

Theorem spsbc 2974
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 1775 and rspsbc 3045. (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 1775 . . . 4  |-  ( A. x ph  ->  [ y  /  x ] ph )
2 sbsbc 2966 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylib 122 . . 3  |-  ( A. x ph  ->  [. y  /  x ]. ph )
4 dfsbcq 2964 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
53, 4imbitrid 154 . 2  |-  ( y  =  A  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
65vtocleg 2808 1  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1351    = wceq 1353   [wsb 1762    e. wcel 2148   [.wsbc 2962
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-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2739  df-sbc 2963
This theorem is referenced by:  spsbcd  2975  sbcth  2976  sbcthdv  2977  sbceqal  3018  sbcimdv  3028  csbiebt  3096  csbexga  4128
  Copyright terms: Public domain W3C validator