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Theorem spsbcd 3175
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 2092 and rspsbc 3240. (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 3174 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
41, 2, 3sylc 59 1  |-  ( ph  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1726   [.wsbc 3162
This theorem is referenced by:  ovmpt2dxf  6200  ex-natded9.26  21728
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-v 2959  df-sbc 3163
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