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Theorem sbcth 3444
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 1720 . 2 𝑥𝜑
3 spsbc 3442 . 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 1479  wcel 1988  [wsbc 3429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1484  df-ex 1703  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-v 3197  df-sbc 3430
This theorem is referenced by:  iota4an  5858  tfinds2  7048  wunnat  16597  catcfuccl  16740  dprdval  18383  bj-sbceqgALT  32872  f1omptsnlem  33154  mptsnunlem  33156  topdifinffinlem  33166  relowlpssretop  33183  cdlemk35s  36044  cdlemk39s  36046  cdlemk42  36048  frege92  38069
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