MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spsbcd Unicode version

Theorem spsbcd 3004
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 1964 and rspsbc 3069. (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 3003 . 2  |-  ( A  e.  V  ->  ( A. x ps  ->  [. A  /  x ]. ps )
)
41, 2, 3sylc 56 1  |-  ( ph  ->  [. A  /  x ]. ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  ex-natded9.26  20806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-v 2790  df-sbc 2992
  Copyright terms: Public domain W3C validator