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Theorem sbcth 2968
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 1442 . 2  |-  A. x ph
3 spsbc 2966 . 2  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
42, 3mpi 15 1  |-  ( A  e.  V  ->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1346    e. wcel 2141   [.wsbc 2955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-sbc 2956
This theorem is referenced by:  rabrsndc  3651  iota4an  5179
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