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Theorem sbcth 3018
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 1536 . 2  |-  A. x ph
3 spsbc 3016 . 2  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
42, 3mpi 16 1  |-  ( A  e.  V  ->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  tfinds2  4670  iota4an  5254  wunnat  13846  catcfuccl  13957  dprdval  15254  cdlemk35s  31748  cdlemk39s  31750  cdlemk42  31752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803  df-sbc 3005
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