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Theorem spsbc 2891
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 1731 and rspsbc 2961. (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 1731 . . . 4  |-  ( A. x ph  ->  [ y  /  x ] ph )
2 sbsbc 2884 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylib 121 . . 3  |-  ( A. x ph  ->  [. y  /  x ]. ph )
4 dfsbcq 2882 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
53, 4syl5ib 153 . 2  |-  ( y  =  A  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
65vtocleg 2729 1  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1312    = wceq 1314    e. wcel 1463   [wsb 1718   [.wsbc 2880
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-v 2660  df-sbc 2881
This theorem is referenced by:  spsbcd  2892  sbcth  2893  sbcthdv  2894  sbceqal  2934  sbcimdv  2944  csbiebt  3007  csbexga  4024
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