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Theorem sbcth 3184
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 1556 . 2  |-  A. x ph
3 spsbc 3182 . 2  |-  ( A  e.  V  ->  ( A. x ph  ->  [. A  /  x ]. ph )
)
42, 3mpi 17 1  |-  ( A  e.  V  ->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1550    e. wcel 1728   [.wsbc 3170
This theorem is referenced by:  rabrsn  3903  tfinds2  4878  iota4an  5472  wunnat  14191  catcfuccl  14302  dprdval  15599  cdlemk35s  31908  cdlemk39s  31910  cdlemk42  31912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-v 2967  df-sbc 3171
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