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

Theorem spsbc 3133
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 2073 and rspsbc 3199. (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 2073 . . . 4  |-  ( A. x ph  ->  [ y  /  x ] ph )
2 sbsbc 3125 . . . 4  |-  ( [ y  /  x ] ph 
<-> 
[. y  /  x ]. ph )
31, 2sylib 189 . . 3  |-  ( A. x ph  ->  [. y  /  x ]. ph )
4 dfsbcq 3123 . . 3  |-  ( y  =  A  ->  ( [. y  /  x ]. ph  <->  [. A  /  x ]. ph ) )
53, 4syl5ib 211 . 2  |-  ( y  =  A  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
65vtocleg 2982 1  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    = wceq 1649   [wsb 1655    e. wcel 1721   [.wsbc 3121
This theorem is referenced by:  spsbcd  3134  sbcth  3135  sbcthdv  3136  sbceqal  3172  sbcimdv  3182  csbexg  3221  csbiebt  3247  pm14.18  27496  sbcbi  28335  onfrALTlem3  28341  csbeq2g  28347  sbc3orgVD  28672  sbcbiVD  28697  csbingVD  28705  onfrALTlem3VD  28708  csbeq2gVD  28713  csbunigVD  28719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-v 2918  df-sbc 3122
  Copyright terms: Public domain W3C validator