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Theorem sbcth 2950
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 1429 . 2  |-  A. x ph
3 spsbc 2948 . 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 1333    e. wcel 2128   [.wsbc 2937
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-v 2714  df-sbc 2938
This theorem is referenced by:  rabrsndc  3628  iota4an  5155
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