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Theorem sbcth 3602
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 1870 . 2 𝑥𝜑
3 spsbc 3600 . 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 1629  wcel 2145  [wsbc 3587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-tru 1634  df-ex 1853  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588
This theorem is referenced by:  iota4an  6013  tfinds2  7210  wunnat  16823  catcfuccl  16966  dprdval  18610  bj-sbceqgALT  33226  f1omptsnlem  33520  mptsnunlem  33522  topdifinffinlem  33532  relowlpssretop  33549  cnfinltrel  33578  cdlemk35s  36746  cdlemk39s  36748  cdlemk42  36750  frege92  38775
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