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Theorem sbcth 3339
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 1596 . 2  |-  A. x ph
3 spsbc 3337 . 2  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
42, 3mpi 17 1  |-  ( A  e.  V  ->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1372    e. wcel 1762   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-v 3108  df-sbc 3325
This theorem is referenced by:  iota4an  5561  tfinds2  6669  wunnat  15172  catcfuccl  15283  dprdval  16818  dprdvalOLD  16820  bj-sbceqgALT  33425  cdlemk35s  35608  cdlemk39s  35610  cdlemk42  35612
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