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Theorem sbcth 2800
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 1354 . 2  |-  A. x ph
3 spsbc 2798 . 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 1257    e. wcel 1409   [.wsbc 2787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576  df-sbc 2788
This theorem is referenced by:  rabrsndc  3466  iota4an  4914
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