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Theorem sbcth 3676
Description: A substitution into a theorem remains true (when 𝐴 is a set). (Contributed by NM, 5-Nov-2005.)
Hypothesis
Ref Expression
sbcth.1 𝜑
Assertion
Ref Expression
sbcth (𝐴𝑉[𝐴 / 𝑥]𝜑)

Proof of Theorem sbcth
StepHypRef Expression
1 sbcth.1 . . 3 𝜑
21ax-gen 1896 . 2 𝑥𝜑
3 spsbc 3674 . 2 (𝐴𝑉 → (∀𝑥𝜑[𝐴 / 𝑥]𝜑))
42, 3mpi 20 1 (𝐴𝑉[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1656  wcel 2166  [wsbc 3661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-13 2390  ax-ext 2802
This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-v 3415  df-sbc 3662
This theorem is referenced by:  iota4an  6104  tfinds2  7323  wunnat  16967  catcfuccl  17110  dprdval  18755  bj-sbceqgALT  33417  f1omptsnlem  33728  mptsnunlem  33730  topdifinffinlem  33739  relowlpssretop  33756  cnfinltrel  33785  cdlemk35s  37011  cdlemk39s  37013  cdlemk42  37015  frege92  39088
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