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Theorem spsbcd 2894
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 Mario Carneiro, 9-Feb-2017.)
Hypotheses
Ref Expression
spsbcd.1  |-  ( ph  ->  A  e.  V )
spsbcd.2  |-  ( ph  ->  A. x ps )
Assertion
Ref Expression
spsbcd  |-  ( ph  ->  [. A  /  x ]. ps )

Proof of Theorem spsbcd
StepHypRef Expression
1 spsbcd.1 . 2  |-  ( ph  ->  A  e.  V )
2 spsbcd.2 . 2  |-  ( ph  ->  A. x ps )
3 spsbc 2893 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
41, 2, 3sylc 62 1  |-  ( ph  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314    e. wcel 1465   [.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:  ovmpodxf  5864
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