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Theorem sbcth 3294
Description: A substitution into a theorem remains true (when  A is a set). (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcth.1  |-  ph
Assertion
Ref Expression
sbcth  |-  ( A  e.  V  ->  [. A  /  x ]. ph )

Proof of Theorem sbcth
StepHypRef Expression
1 sbcth.1 . . 3  |-  ph
21ax-gen 1641 . 2  |-  A. x ph
3 spsbc 3292 . 2  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
42, 3mpi 21 1  |-  ( A  e.  V  ->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1405    e. wcel 1844   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-v 3063  df-sbc 3280
This theorem is referenced by:  iota4an  5554  tfinds2  6683  wunnat  15571  catcfuccl  15714  dprdval  17356  dprdvalOLD  17358  bj-sbceqgALT  31046  f1omptsnlem  31265  mptsnunlem  31267  topdifinffinlem  31277  relowlpssretop  31294  cdlemk35s  33969  cdlemk39s  33971  cdlemk42  33973  frege92  35949
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