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Theorem spsbc 2851
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 1705 and rspsbc 2921. (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 1705 . . . 4  |-  ( A. x ph  ->  [ y  /  x ] ph )
2 sbsbc 2844 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylib 120 . . 3  |-  ( A. x ph  ->  [. y  /  x ]. ph )
4 dfsbcq 2842 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
53, 4syl5ib 152 . 2  |-  ( y  =  A  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
65vtocleg 2690 1  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287    = wceq 1289    e. wcel 1438   [wsb 1692   [.wsbc 2840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621  df-sbc 2841
This theorem is referenced by:  spsbcd  2852  sbcth  2853  sbcthdv  2854  sbceqal  2894  sbcimdv  2904  csbiebt  2967  csbexga  3967
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