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Theorem sbcth 3785
 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 1789 . 2 𝑥𝜑
3 spsbc 3783 . 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 1528   ∈ wcel 2107  [wsbc 3770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-sbc 3771 This theorem is referenced by:  iota4an  6330  tfinds2  7570  wunnat  17218  catcfuccl  17361  dprdval  19117  opsbc2ie  30231  bj-sbceqgALT  34207  f1omptsnlem  34604  mptsnunlem  34606  topdifinffinlem  34615  relowlpssretop  34632  cdlemk35s  38060  cdlemk39s  38062  cdlemk42  38064  frege92  40286
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